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ANALYTIC GEOMETRY 




A FIRST COURSE 



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ANALYTIC GEOMETRY 



A FIRST COURSE 



BY 



WILLIAM H. MALTBIE 
n 



1906 

THE, SUN JOB PRINTING OFFICE 
BALTIMORE 



2& 



LIBRARY of CONGRESS 
Two Copies Received 
OCT 19 1906 
^ Cepyrlf ht Entry 

CUSS CL XXC, No 

COPY B. 



COPYRIGHT, 1906 

BY 

WILLIAM H. MALTBIE 



TABLE OF CONTENTS 



Article. Page. 

1, 2 Introduction 3 

ANALYTIC GEOMETRY OF ONE DIMENSION. 

3 The system of co-ordinates 5 

4 The relation of the point and the equation 6 

5 The equation with equal roots 7 

6 The equation with complex roots 8 

7 Transformation of co-ordinates in one dimension 9 

8 The derivation of the formula of transformation which will 

produce a given result 11 

9 Distance ratios and anharmonic ratios 12 

SOME FUNDAMENTAL IDEAS OF THE ANALYTIC GEOMETRY OF TWO DIMENSIONS. 

10 The system of co-ordinates 14 

11 The significance of two simultaneous equations 16 

12 The significance of a single equation. Loci 17 

TO CONSTRUCT THE LOCUS WHEN THE EQUATION IS GIVEN. 

13 The process of curve plotting 19 

14 The utility of the process 22 

TO CONSTRUCT THE LOCUS WHEN A FINITE NUMBER OF POINTS UPON IT IS 

GIVEN. 

15 The method of plotting 24 

16 Choice of scales of measurement 25 

17 Determination of intermediate values 26 

18 Definition and test of continuity 27 

19 Infinite discontinuity 28 

TO DEDUCE THE EQUATION WHEN THE RESTRICTIONS ON THE MOVEMENT OF 
THE TRACING POINT ARE GIVEN. 

20 The general method 29 

21 Distance between two points in terms of their co-ordinates. ... 30 

22 Co-ordinates of a point with a given distance ratio 30 

23 Angle between a line and the X axis .32 

24 Illustrations of the development of equations of loci 32 

TO DEDUCE THE EQUATION WHEN A FINITE NUMBER OF POINTS ON A LOCUS 
OF SOME KNOWN TYPE IS GIVEN. 

25 Limitations of the problem. Outline of the method 35 

26 Number of points required to determine a curve 36 

27 Complex conditions equivalent to two or more simple ones. ... 37 



Table of Contents. . 

TRANSFORMATION OF CO-ORDINATES. 

Article. Page. 

28 The general problem 38 

29 Movement of the axes parallel to themselves 38 

30 Rotation of the axes 40 

31 Change of the angle between the axes 42 

32 Change of the scale of measurement 43 

33 The general transformation 43 

34 Transformations interpreted as changes of the loci 43 

INTERSECTION OF LOCI. 

35 The significance of imaginaries 45 

36 Possibility of error 46 

THE EQUATION OF THE FIRST DEGREE AND THE STRAIGHT LINE. 

37 Standard forms of the equation of the straight line 48 

38 The equation of the straight line through two given points. ... 51 

39 The straight line given by two equations in three variables. ... 52 

40 Distance from a point to a line 53 

41 Normal form of the equation of a straight line. 54 

42 Intersections of lines 56 

43 Families of lines 57 

44 The line at infinity 58 

45 Parallel lines 59 

THE CIRCLE, A SPECIAL CASE OF THE EQUATION OF THE SECOND DEGREE. 

46 The general equation of the circle 61 

47 Intersections of circles 62 

48 Tangents and normals . .: 63 

49 Condition of tangency 64 

50 Equation of the tangent through a given point 65 

51 Sub-tangent and sub-normal 67 

52 Poles and polars defined , 68 

53 Equation of the polar 69 

54 Co-ordinates of the pole 70 

55 Polar as locus of harmonic conjugates 70 

56 Polar as locus of poles 71 

ADDITIONAL WORK ON THE SUBJECT OF LOCI. 

57 General remarks on loci problems 74 

THE GENERAL EQUATION OF THE SECOND DEGREE. 

58 Nature of the problem and of the method employed 78 

59 The "r" equation ' 78 

60 One chord is bisected at any point 79 

61 Center of a conic 79 

62 Diameters of a conic 80 

63 Axes of symmetry of a conic 80 

64 Reduction of the general equation 81 

THE ELLIPSE AND THE HYPERBOLA. 

65 Determination of form 84 

66 Early geometric definitions 86 

67 Mechanical constructions 91 



Table of Contexts. 

Article. Page. 

68 Diameters 93 

69 Supplemental chords 96 

70 Tangents and normals 97 

71 Poles and polars 99 

72 Asymptotes 99 

73 The conjugate hyperbola 101 

74 The auxiliary circles 102 

THE PARABOLA. 

75 Determination of form 104 

76 Early geometric definition 104 

77 Mechanical construction 105 

78 Diameters 106 

79 Tangents and normals 107 

80 Poles and polars : 108 

ADDITIONAL WORK OX THE GENERAL EQUATION OF THE SECOND DEGREE. 

81 Necessity of a general treatment. 109 

82 Degenerate conies 110 

83 Discriminant Ill 

84 Classification 112 

85 Tangents and normals 112 

86 A second condition of tangency 114 

87 Pole and polar 116 

88 Length of axes 117 

89 Foci and eccentricity 117 

90 Asymptotes 117 

91 Special treatment for the parabola 119 

92 Higher Loci 122 

93 Families of conies 122 

OTHER SYSTEMS OF CO-ORDINATES. POLAR CO-ORDINATES. 

94 Various systems of co-ordinates 123 

95 Merits and demerits of various systems 124 

96 Polar co-ordinates 125 

97 The relation of polar and cartesian co-ordinates 127 

APPENDICES. 

A Infinities of various orders 129 

B Functionality 131 

C Permissible operations 133 

D Projection 135 

E Imaginaries 138 



ANALYTIC GEOMETRY. 

CHAPTER I. 

Introductory. 

1. The student who begins the study of Analytic Geometry is 
assumed to have acquired previously a knowledge of elementary 
geometry and of algebra. 

In geometry he has used as subject matter the geometric ele- 
ments (point, line, plane, etc.) and, starting from certain well 
defined axioms and postulates, has deduced the properties of the 
simpler plane and solid geometric forms. He has also developed 
a method of investigation which enables him to deal with questions, 
of form, and of magnitude so far as it depends on form. In algebra 
he has used as subject matter certain symbols, some quantitative 
{%. y, a, . . . . ), some operational (+, — , =, V? ^-> • •) 5 an( ^ 
has developed a method, far more general than that of arithmetic, 
of dealing with questions of quantity. 

Each of these methods of mathematical investigation (geometric 
and algebraic) has its advantages and its limitations. In geom- 
etry we work with elements which actually possess the properties 
of form, position, and magnitude in Avhich we are interested, and 
the method of proof is in consequence' frequently suggested by a 
glance at the diagram. In algebra we work with symbols which 
have in themselves no properties and carry no power of suggestion. 
Consider the two statements 

a. Two straight lines cannot intersect in more than one point ; 

b. The equations 2% — 3i/ + 1 = 
and ±x + By — 3 = 

cannot be simultaneously satisfied for more than one pair 

of values of the variables. 
These statements as we shall see later are practically one and 
the same ; but the first is evident as soon as the diagram is drawn, 
while the second requires a proof of whose form 'the equations 
themselves give no hint. The advantage of the geometric method 
is evident. 



s 



4 Analytic Geometry. 

The algebraic method on the other hand possesses a great ad- 
vantage over the geometric in the remarkable generality of its 
processes and results. It includes whole classes of problems in a 
single equation and expresses the solution of them all in a single 
formula. For example all quadratics may be included in the 
single form 

ax 2 -f- Zbx -\- c = Q, 

and the solutions of them all are embraced in a single formula, 

b 1 



* = -- ±-Vo-ac. 

2. Analytic Geometry is an attempt to establish a relation 
between these two branches of mathematics, so that the methods 
of either may be applied to the other; in short, an attempt to 
•establish such a connection that one may write the formula of a 
•curve or draw the diagram of an equation. 

At the very outset we are confronted by a difficulty. Geometry 
deals almost wholly with fixed objects, definite points, lines, and 
planes, while algebra is concerned largely with variables. But we 
can obviate this difficulty by thinking from now on of all curves 
as traced out by a variable point, whose variation* in position we 
shall attempt to connect with the change in value of algebraic 
variables. 



CHAPTER II. 

ANALYTIC GEOMETRY OF ONE DIMENSION. 

3 - Consider the simplest case of algebraic 

THE SYSTEM OF variation. A single variable x is free to take 
co-ordinates. a n possible values from minus infinity to plus 

infinity. An equation of the first degree, 
x — a == 0, 

stops the variation of x and compels it to take the value a. An 
equation of the second degree, 

(x — a) (x — 1)) = x 2 — (a J r b)x J r ab = 0* 

compels x to take one of the two values a or h; and similarly for 
the equations of the higher degrees. Compare all this with the 
simple case of geometric variation by which a point traces otlt a 
straight line. The point takes an infinite number of different 
positions, between which and the values of x we may establish k 
one to one correspondence. But in order to do this we must make 
certain purely arbitrary assumptions. 

A. We must assume that a certain point on the line is to repre- 
sent the zero value of x. This point we shall call the origin. Let 
it be the point 0. 

? 

Fig. 1. 

B. In order to represent both the positive and negative values 
of x we must divide our line into positive and negative portions. 
We assume that all points on the right of the origin correspond to 
positive values of x, and all points on the left to negative values. 

C. We must decide upon a scale of measurement ; i. e., whether 
we will measure distances on the line in feet, inches, millimeters, 
or by some other arbitrary scale. 



*The symbol = is used to denote an identity. The student must be 
thoroughly familiar with the distinction between conditional and identical 
equations. 



6 Analytic Geometry. 

Having made these three assumptions, we agree that the point 
at the distance a (measured by the agreed upon scale) from the 
origin shall correspond to the value a of x, and the desired corre- 
spondence between the point and the variable is now established. 
As x changes continuously from minus infinity to plus infinity, 
the point, by virtue of the assumptions we have made, is forced to 
move continuously from one end of the line to the other, while to 
any one value of x there corresponds one and only one position of 
the point and conversely. 

The distance of a point from the origin is called the co-ordinate 
of the point, while the line which is used as the basis of the 
system is called the axis of co-ordinates, or merely the axis. 

4. If a point is at the distance a from the 

THE RELATION OF origin the equation 
THE point AND x = a,ovx-a = 0, 

THE EQUATION. . T . , ,. ,. _ ,_ , . , _. .. 

is called the equation of that point. Thus if 

the centimeter is the unit of measure the 
point P in Fig. 1 is spoken of as the point 4, and its equation is 

x — 4 = 0. 
It is at once evident that every first degree equation in one variable 
represents a single point, and conversely that any point on the line 
may be represented by a first degree equation in one variable. 

The second degree equation in one variable can be written as 
the product of two factors, and is satisfied when either of the 
two factors is equal to zero. It is accordingly said to represent 
the two points which would be represented by the two factors 
taken singly. For example, 

x 2 — 3x + 2 = (x — 2) (x — 1 ) == 
represents the two points at distances two and one to the right 
of the origin. 

PROBLEMS. 

Locate the points represented by the following equations : 
1. 3# = 4 2. x+ f ^ = 5 

5. x 2 — 5x + 6 = 6. 2x 2 — 7x — 2 = 

7. x(x 2 + Ax — 12)= 8. (2x 2 — lOx — 12) {x 2 — 1) = 

9. x 2 — l = 7x — S 10. x 3 — 4a? = 



Analytic Geometry. 7 

11. .Write the equation which represents two points on opposite 
sides of the origin and at a distance five from it. 

12. If the unit of length adopted be the foot, write the equations 
of the points whose distances from the origin are respectively 3 
inches, 3 feet, 3 yards, minus 1 yard. 

13. Write the equation representing the three points whose co- 

1 4 
ordinates are a, — , — . 

2 3 

14. Generalize the work in problem 11 ; i. e., write the equation 
which, if proper values are given to the constants involved, will 
represent any pair of points symmetrically situated with respect to 
the origin. 

15. Write the general equation representing a triad of points 
whose co-ordinates are in the ratio of 2, 3, 4. 

5. The equation 

THE EQUATION 



WITH EQUAL 
ROOTS. 



x 2 — 2ax + a 2 = 

demands special consideration. It reduces 
at once to 

(x — a) (x — a) = (x — a) 2 = 0, 



and we might say that the equation represents only the point at 
the distance a from the origin. But this mode of interpretation 
is unwise because it fails to recognize any distinction between the 
two equations 

x 2 — 2ax + a 2 = 
and x — a = 0. 

We shall obtain a hint of the proper interpretation if we ask 
how the equation under consideration arises. It is evidently the 
limiting form of 

(x — a) (x — ~b) = x 2 — (a -\- b) x -\- al) = 

as 1) tends to a as a limit. W r e may therefore say, 

a x 2 — 2ax + a 2 = 

is the limiting form of an equation representing two distinct 
points as the points tend to coincidence." Mathematicians, how- 
ever, are accustomed to use the shorter expression, 

«x 2 — 2ax + a 2 = 

represents two coincident points." understanding by this exactly 
what is expressed in the longer phrase above. 



8 Analytic Geometry. 

The method of interpretation adopted here for this exceptional 
case of the second degree equation is a general method, widely 
used in all branches of mathematics, and the student should 
obtain a clear understanding of it. It may be stated as follows : 

1. Exceptional cases will be treated by regarding them as limit- 
ing forms of the more general case. 2. If any expression has been 
used to represent the more general case, the limiting form of that 
expression will be used, whenever it is possible, to represent the 
special case.* 

6. Another type of equation presents a more 

THE EQUATION serious difficulty. When tf — ac is negative 

WITH complex the equation 

ROOTS.t ax 2 + 2lx + c = 

is not satisfied by any real value of x. Con- 
sider the special case of 

# 2 — 2^4-2 = 0. 



Here x = 1 ± V — 1 

and there are no points on the line corresponding to these values 
of x. The difficulty grows out of the nature of our fundamental 
assumptions. We have established such a connection between the 
points on the line and the values of x that every point corresponds 
to a real value of x and there are no points left to correspond to 
imaginary or complex values. We shall accordingly say that the 
equation represents a pair of imaginary points, meaning thereby 
merely that it represents a pair of points that from the nature 
of our fundamental assumptions cannot be represented in our 
diagram. 



*As another illustration consider the treatment of the special case of 
division by zero. It is regarded as the limiting form of division by a 
quantity b, as b tends to zero. It is evident, provided the dividend is not 
zero, that as the divisor tends to zero the quotient increases indefinitely. 

Our general expression for division is — == c, and in accordance with the 

b 

second part of our principle we write — = oo . This expression must not, 

however, be regarded as containing any statement as to the possibility of 
dividing something by nothing. It is merely a short hand way of saying, 
''When the dividend is not zero and the divisor tends to zero as a limit 
the quotient increases indefinitely." Other examples of this mode of inter- 
pretation will be met with from time to time in our work. 

tA complex number is one of the form a + b V — 1 . When a is zero 
the number is a pure imaginary, when b is zero it is real. From now on 
we shall denote V — 1 by i and the general form of a complex number will 
be a 4- ib. 



Analytic Geometry. 9 

PROBLEMS. 

What points do the following equations represent? 

1. a? 2 — 3a? — 28=0 2. 3a? 2 — 20 + 1 = 

3. (a? 2 — 4a? + 9) (a? — 2) = 4. (a? 2 — 1) (a? 2 + 1)=0 

5. a' 2 (a? 2 — 3a? — -£- )=0 0. a? (a? 2 — 4a? + 1) (a? 2 + 2) = 0. 

4 

7- In building our system of co-ordinates we 

transformation made three purely arbitrary assumptions: 
OF co-ordinates nrs t ? that the origin was at a particular 
IN one dimension, point; second, that distances to the right 
were to be counted positive; third, that dis- 
tances were to be measured by a given scale. A change in any 
one of these assumptions will of course lead to a new system of 
co-ordinates, and the algebraic relation between the two systems 
must be determined. 

p R. a 

Fig. 2. 

Suppose for example that the point P has been taken. as the 
origin, distances to the right as positive, and — of an inch as the 

unit of measurement. The point Q will then have the co-ordinate 
10 and be represented by the equation 

a? — 10=0. 

If now we take as a new origin a point R at a distance 4 to the 
right of P, and, to avoid confusion, call the variable in this new 
system a?', we see at once that the old and new co-ordinates of any 
point are connected by the relation 

x = x + 4, 

and that the new equation of. Q is therefore 

a?' — 6 = 0. 

In general a movement of the origin a distance h, positive or 
negative, corresponds to the algebraic substitution 

x = x + h. 

If we change our assumption as to the directions and make 
distances to the left positive, it is evident that the corresponding 
substitution is 

x== — a?'. 



10 Analytic Geometry. 

If we change our assumption as to the unit of length, and re- 
place it by one k times as great, it is evident that the correspond- 
ing algebraic substitution is 

x = kx . 

If we note that the substitution 

x = x + h 
followed by x = — x" 

followed by x" = kx" 

gives us x = — kx" + h. 

we see that the substitution 

x = ax' + 1) 

may be made by proper choice of the constants a and ft to represent 
any change whatever of the system of co-ordinates. Any such 
change is called a transformation of co-ordinates and the corre- 
sponding algebraic substitution is called a formula of trans- 
formation. 

The mathematical importance of the subject of transformation 
of co-ordinates grows out of the power which the corresponding 
substitutions give us to modify equations. For example, the sub- 
stitution 

x = x + 2 





reduces 


X 2 — 4# + 3 = 


to the simpler form 


x 2 — 1 = 0, 


while the substitution 






X = x + 1 



reduces the same equation to 

x' 2 —2x=0. 
Again the equation 

9x 2 — 6x — 8 = 

which has the fractional roots - and — - is reduced by the 

3 3 

x' 
substitution x = — 

3 
to the form x 2 + 2a?' — 8 = 

which has the integer roots 4 and 2. 



Analytic Geometry. II 

PROBLEMS. 

Explain the geometric significance of the following transfor- 
mations. 
1. x = qb'-\-2 2. w — x — 4 

3. 2# = 2#' + 3 ■ 4. x = —2x 

5. x = — 3a?' — 4 6. 2x=5 — 6x 

Write the formulae of transformation which shall represent the 
following changes of the system of co-ordinates : 

7. A movement of the origin a distance 4 to the left. 

8. A movement of the origin to the point whose co-ordinate is 
_ 3_ 

2 ' 

9. A division of the unit of length by 4. 

10. A movement of the origin a distance 2 to the right, followed 
by an interchange of the positive and negative portions of the 
axis. 

11. A movement of the origin to the point whose co-ordinate is 
4, followed by a multiplication of the unit of length by three, fol- 
lowed by an interchange of the positive and negative portions of 
the axis. 

8 - We are not always given the formula of 

the derivation OF transformation. Cases frequently ar*se in 
the formula OF which we are required to determine the sub- 
transformation stitution which will produce .a given form 
WHICH WILL when applied to a particular equation. For 

produce A given example, let it be required to find the trans- 
result. formation that will reduce the equation 

x 2 — 5^ + 6=0 

to a new equation which has no term of the first degree in the 
variable. Two methods of solution present themselves. We may 
find the two points represented by the equation and then ac- 
complish the desired result by taking the point midway between 
them as the new origin, (Prob. 14, Art. 4) ; or we may assume a 
general substitution 

x == x '-J-- K 

and note that the resulting equation 

x' 2 + x'(2h — 5) + ft 2 — 5ft + 6 = 
has no term of the first degree in the variable if 

2ft— 5=0, i. e., if ft= A 

2 
Our substitution is therefore determined. 



12 Analytic Geometry. 

The second of the methods used is the more general and there- 
fore of course the more valuable. 

PROBLEMS. 

Use each of the methods outlined above to determine the sub- 
stitutions which will reduce the following equations to equations 
having no term of the first degree in the variable : 
1. a? 2 + 3a? — 7 = 2. a? 2 — 4a? — 1 = 

3. x 2 — 5% + 2=0 4. x 2 — 2x + 1 = 0. 

5. Discuss in full the substitution which will reduce an equation 
of the second degree to an equation wmich has no constant term. 

6. Show that no movement of the origin can change the degree of 
the equation. 

9. If we are given tw r o points A and B, whose 

distance ratios co-ordinates are x, and x 2 , and a third point 

AND ANHARMONIC 
RATIOS. 



in the ratio 



C, whose co-ordinate is x 3 , on the same line, 
the point C is said to divide the segment AB 

which is called the distance ratio of C with respect to A and B. 

This distance ratio is evidently numerically equal to the ratio of 

A C 
the segments — — and is positive or negative according as the 

BC 
point C is within or without the segment AB. In this latter fact 
lies its superiority to the mere numerical ratio, which makes no 
distinction between internal and external division. 

If a fourth point D whose co-ordinate is x 4 is given , the ratio 
of the two distance ratios so determined 

x x x s 



®1 



X 4 X<> 

i. e., 

(x 1 — x 3 )(x± — x 2 ) 

(x ± a? 4 ) {x 3 x 2 ) 

is called the cross ratio or the anharmonic ratio of the four points, 
and is denoted by 



Analytic Geometry. 13 

If C and I) divide the segmenl AB internally and externally in the 
same ratio, the anharmonic ratio is — 1, the division is said to 
be harmonic, and C and D are said to be harmonic conjugates with 
respect to A and B. 

PROBLEMS. 

1. Find the distance ratio of the point 4 with respect to the 
points 3 and 7. 

2. Find the anharmonic ratio of the points 2 and — 1 with 
respect to the points 3 and 5. 

3. Find the point whose distance ratio with respect to the 
points 2 and 4 is - — 5. 

4. Find a point x such that the anharmonic ratio of 7 and x with 
respect to 4 and 5 shall be 2. 

5. Find the harmonic conjugate of the origin with respect to 
4 and — 1. 

G. Generalize problem 3, i. e., find the point whose distance ratio 
with respect to x x and x 2 shall be k. 

7. Show that the definition given above of harmonic division is 
equivalent to the following: The segment AB is divided harmon- 
ically bv P and Q if — + J— = — . 
0^ QA QP 



CHAPTER III. 

SOME FUNDAMENTAL IDEAS OF THE ANALYTIC 
GEOMETRY OF TWO DIMENSIONS. 

10 - We have built up in the preceding pages 

THE system QF an analytic geometry of one dimension in 

co-ordinates. order to illustrate the variation of a single 

algebraic variable. Let us now consider the 

less simple case of two variables. 

Two variables, x and y, are each free to take any value from 
minus infinity to plus infinity and these values are paired (one of 
w with one of y) in all possible ways. We desire to establish a one 
to one correspondence between these pairs of values and the mem- 
bers of some group of geometric objects, just as we established a 
one to one correspondence between the values of one variable and 
the points of a straight line. Since with any one of the infinite 
number of values of x may be paired any one of the infinite 
number of values of y, the total number of pairs is infinity squared, 
or to use a more satisfactory phrase, is doubly infinite, or an 
infinity of the second order.* The number of points in a line 
however is singly infinite, and our former mode of representation 
therefore fails us, since we cannot establish a one to one corre- 
spondence between a singly infinite number of points and a doubly 
infinite number of pairs of values. The plane however contains 
a doubly infinite number of points and may therefore be used. 

In order to establish this correspondence, between the pairs of 
values of x and y and the points of the plane we must make certain 
arbitrary assumptions. 



"See Appendix A. Infinities of various orders. 




Analytic Geometry. 15 

A. We assume two intersecting straight lines as a basis of 
reference. Let them be OA and OB. 

B. We assume positive and neg- 
ative directions on these two lines. 
Let OA and OB be positive, and OA' 
and OB' negative. 

C. We assume that values of x 
correspond to distances from the line 
BB' measured (by any desired unit 

^ of length) parallel to the line A A' '. 

D. We assume that values of y 
correspond to distances from the line 
A A' measured (by any desired unit 

of length) parallel to the line BB' . (The units determined by the 
assumptions C and D will usually but not always be the same.) 

The lines AA' and BB' are called Axes of Co-ordinates. AA' 
is the X axis or axis of abscissas. BB' is the Y axis or axis of 
ordinates. The intersection of the axes is the origin. The distance 
of a point from the Y axis, measured parallel to the X axis, is 
called the abscissa, or the x co-ordinate, or frequently merely the a? 
of the point. The distance of a point from the X axis, measured 
parallel to the I 7 axis, is called the ordinate, or the y co-ordinate, 
or frequently merely the y of the point. Thus in the diagram the 
point P has the abscissa a, and the ordinate ft. The point P is 
frequently spoken of as the point ( a, h). In this notation the x 
co-ordinate is always the one first mentioned. When the axes are 
perpendicular to each other the co-ordinates are called rectangular 
co-ordinates, otherwise they are oblique co-ordinates. From now 
on we shall understand rectangular co-ordinates to be used unless 
it is otherwise stated.* 

The four assumptions made above establish a complete corre- 
spondence between the points in the plane and all pairs of real 
values of x and y. To each point corresponds one and only one 
pair of values and conversely. 



*Such a system of co-ordinates as we have outlined above' is frequently 
called Cartesian, in honor of Descartes, who was the first worker in this 
field. Consult Ball's Short History of Mathematics or some similar work. 



16 Analytic Geometry. 

PROBLEMS. 

1. Locate the following points: (1, 1), ( — 1, — 1), (0, 3), 
(2, 0), (-5, 8), (0, 0), (5, -l),(-2, 4|), g, -JY (-2, -4), 

(0,-5). 

2. The lower left hand vertex of a square whose side is 4 is at 
the origin and two sides coincide with the axes. Find the co- 
ordinates of the other three vertices. 

3. The upper right hand vertex of a rectangle whose sides are 
8 and 3 is at the origin and its longer side coincides with the X 
axis. Find the co-ordinates of the middle points of the sides. 

4. The base of an equilateral triangle whose side is 4 is parallel 
to the X axis, and the point (0, 2) is the middle point of the base. 
Find the co-ordinates of the three vertices. 

5. The axes are the diagonals of a square whose side is 8. Find 
the co-ordinates of the vertices. 

G. A regular hexagon whose side is 5 has its center at the 
origin and one pair of vertices on the X axis. Find the co- 
ordinates of the remaining vertices. 

11 - If 00 and y are subject to no restriction the 

THE SIGNIFICANCE corresponding point may take any position 

OF TWO i n the plane. If, however, we are given two 

SIMULTANEOUS equations which the variables must satisfy, 

equations. the point is no longer free. For example 

3x — y — 5=0 
and 5a? + y — 11 = 

are both satisfied only wiien we have 

so = 2, y = l. 

These two equations therefore restrict the varying point to the 
single position (2, 1), and may therefore be said to represent this 
point. Again 

® 2 + y 2 — 6a? + 4 = 

and x-\-y = 

may be said to represent the two points (1,1) and (2, 2) since only 
for these values of x and y can these two equations be simulta- 
neously satisfied. 

Since two algebraic equations in two variables always on solu- 
tion give a finite number of pairs of values of the variables, it 
follows that two algebraic equations in two variables restrict the 
varying point to a finite number of fixed positions. 



Analytic Geometry. 17 

PROBLEMS. 

Find the points represented by the following pairs of simulta- 
neous equations: 

1. ±x — t/ + 22 = 2. 3x + y + 2 = 
x — 2y + 9 = 6a? — %y— 11 = 

3. 4a? — 7/ — 4 = 4. a? + 2# = 
5a?+2*/ — 5 = 3a?— 4i/ = 

5. x 2 + y 2 — 9 = 6. 7/ 2 — 8a?=0 
a? +2*/ — 1 = #+2?/ = 

12 - When we have a single equation the values 

THE significance f x an d y are n0 longer determined, but are 
OF A single nevertheless not entirely free. The equation 

equation. LOCI. acts as a restriction on the movement of the 
variable point, without fixing its position. 
For example 

a? — y = 

is satisfied by the points* (1, 1) , (2, 2) , (3, 3) but not by the points 
(1, 2), (1, 4), (2, 3). In fact it is satisfied by every point whose 
co-ordinates are equal and of the same sign and by no others. 
It is therefore satisfied by all points on that bisector of the 
angle between the axis which passes through the first and third 
quadrants, and by no others. This bisector is therefore called the 
locus of points subject to the given condition, and 

x — y = 

is called the equation of the bisector, or the equation of the 
locus. In the same way any equation between x and y acts as a 
restriction on the variation of the point which has x and y as its 
co-ordinates. The aggregate of all points which satisfy a given 
condition is said to constitute a locus, and the equation which 
expresses the condition is called the equation of the locus. The 
student may accept without proof for the present the statement 
that the locus, as used in, analytic geometry of two dimensions, 
consists in general of one or more lines, straight or curved. 

It sometimes happens that the left hand member of the given 
equation is the product of two or more rational factors. In this 
case it is evident that the equation will be satisfied by such 



*This is the usual abbreviated form for the expression, "The equation 
x — i/ = is satisfied by the co-ordinates of the points, ' etc. 



18 Analytic Geometry. 

points as make any one of the factors equal to zero ; i. e., the 
locus in this case consists of two or more parts, each of which 
would be represented by equating one of the factors to zero. 
Such a locus is said to be a degenerate locus. For example 

(x — ij)(x + y)=Q 

is satisfied by all points on the bisector through the first and third 
quadrants, but also by the points on the bisector through the 
second and fourth quadrants. The equation is said therefore to 
represent the degenerate locus consisting of these two bisectors. 

This relation between equation and locus is the fundamental 
idea in our present subject. The problems to which it gives rise 
may usually be divided into the following groups : 

a. To construct the locus when the equation is given. 

b. To construct the locus when a finite number of points upon 
it is given. 

c. To deduce the equation when the restrictions on the move- 
ment of the tracing point are given. 

d. To deduce the equation when a finite number of points on a 
locus of some known type is given. 

e. To deduce the geometric properties and relations of loci from 
a consideration of the corresponding equations. 

These problems we shall now proceed to treat in turn. 



CHAPTER IV. 

TO CONSTRUCT THE LOCUS WHEN THE EQUATION IS 

GIVEN. 

13. It sometimes happens, as in the case dis- 

THE process OF cussed at the opening of the previous para- 
CURVE plotting, graph, that the equation is so simple that 
the form of the locus can be at once inferred 
frcan the equation ; and the ability to recognize in this way 
a large number of simple loci can easily be acquired. 

In more complex cases it is not so easy to infer the form of the 
locus from the equation, and the following method is adopted. 
Assume in succession a number of values of one variable, sub- 
stitute these in the equation and compute the corresponding 
values of the other. Each of the pairs of values thus determined 
will locate a point on the locus. By locating a sufficient number 
of such points the form of the locus may be inferred. For ex- 
ample, consider the equation 

x—Sy + 2 = 0. 

If we substitute in succession for x in this equation the integer 
values from — 7 to + 7, and compute the corresponding values 
of y, we obtain -the following points which satisfy the equation : 

(-^-f), <-«. --■§), (-5,-1), (-4,-f ),(-3,-|), 

(-2,0), (-1, 1), (0, J), (1, 1), (2, |), (3, |), (4, 2), 

(5, J), (6, |), (7, 3). 

If we locate these points on our diagram we shall find that they 
all lie on a straight line, and consequently we may infer that this 
line is the locus. But in drawing this inference we make two 
assumptions: first, that the point which traces the locus moves 
continuously and not by leaps from point to point, or in other 



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Analytic Geometry. 



21 



words that the locus is a continuous curve and not a number of 
separate fragments or isolated points; second, that the tracing 
point moves along the curve AB 
rather than along some other 



curve through the same points, 
as the undulating curve in the 
figure. 

We can practically convince 
ourselves of our right to make 
these assumptions in this par- 
ticular case by taking values of 
x between those already taken, 
and in this way finding addi- 
tional points of the locus. 
The mathematical treatment of 
these difficulties must be de- 
ferred to a later point in the student's career. 

As a second example, consider the equation 
Solve the equation for y and we have 



Fig. 4. 



y = ± j V9 — x\ 

We see at once that it is useless to look for points whose x does 
not lie between — 3 and + 3, since any value of x outside these 
limits gives y imaginary values, and our fundamental assumptions 
are such that a pair of values, one ok both of which are imaginary, 
has no corresponding point in the plane. Giving x a series of 
values between — 3 and + .3, and computing the corresponding 
values of y, w T e obtain the following points which satisfy the 
equation 

C-8,-0),.(-a,±fyfi), (-1, +I-T/2), (0, ±2), (1, ±| i/2), 



(2, ±|V5),(3, 0) 



22 



Analytic Geometry. 




Fig. 



Plotting these points we readily 
infer the curve to be of the form iii 
the figure, an inference which we 
may confirm as in the last example. 
It is sometimes more convenient 
to assume arbitrary values for y and 
compute the corresponding values 
of x, as for example in the equation 

x = y 2 — 3i/ + 2. 
The variable to which arbitrary val- 
ues are assigned is called the independent variable, the other 
the dependent variable. The distinction is evidently a purely 
arbitrary one and the student Avill in each case so choose the 
independent variable as to make his work as simple as possible. 
14. The process illustrated in the last para- 

UTILITY OF THE graph, and usually spoken of as curve plot- 
p ROC ESS. ting, frequently enables us to visualize a 

formula and obtain a clearer idea of its sig- 
nificance than we could obtain in any other manner. For 
example, consider the case of a body of mass m moving with a 
velocity v. Its momentum 21 and its kinetic energy E are given by 
the formulae M = mv 

and E =—mv 2 . 

2 

A study of these formulae will give the student some idea of 
the relative variation of momentum and energy as the velocity 
changes, but a much clearer idea will be obtained by an examin- 
ation of Fig. 6, where the corresponding curves are plotted on the 
same axes. In each case the horizontal axis is chosen as the 
axis of velocities, and in order to make the problem a definite 
one the mass m is taken as unity. 

PKOBLEMS. 

Plot the following curves, inferring, whenever you can, the 
form of the curve directlv from the equation.* 

1. # = 2. y = 

3. y =2 ' 4. 3x = 6 

5. x -\-y = 6. x — 2y = 

7. 2a? + 4y = 8. x 2 +y 2 =± 

9. # 2 + 2?/ = 10. x 2 +y 2 — 2x = 



11. x + y — l = 



12. y = x 2 + 3x — 1 



*The student should use co-ordinate paper ruled to tenths. 



Analytic Geometry. 23 

L3. The distance passed over by a moving body in t seconds 
is given by the formula 

d= vj + — atf y 

where /•„ is the initial velocity and a is the acceleration. Plot the 
curves for the three special cases 

(1) v — 0, a = 2, 

(•2) v = 2, a = 0, 

(3) v = 2, a = 2, 

and compare them. 

14. The intensity of the light at any point varies inversely as the 
square of the distance of that point from the source of light; i. e., 

i — -=j. Let k equal one. and plot the corresponding curve. 

15. The intensity of the magnetic field in the vicinity of a wire 
carrying an electric current varies inversely as the distance from 

the wire; i. e., i= -—-. Let fc=l, plot the corresponding curve, 

and compare it with the results of the last problem. 

16. Show that the locus represented by the equation 

ax -\- by -f- c = 

can have no point in the first quadrant if a, ~b, and c are all 
positive. 

1 7. State a corresponding theorem for the third quadrant. 



CHAPTER V. 

TO CONSTRUCT THE LOCUS WHEN A FINITE NUMBER 
OF POINTS UPON IT IS GIVEN. 



15. 



THE METHOD OF 
PLOTTING. 



The equation of the locus is not always 
given. The law which regulates the phe- 
nomena may not be known or may be too 
complex for simple mathematical expression. 
We may however still have sufficient data to plot the 
curve. For example, a student in the laboratory desires 
to study the rate at which a body cools. He may not know the 
formula which connects the temperature of the body with the 
length of time it has been allowed to cool; but he records the 
temperature at short intervals and secures in this way a number of 
pairs of values of time and temperature which he may regard as 
co-ordinates, and thus plot a number of points on the curve. This 
curve (Fig. 7, p. 20) shows clearly the relation between temper- 
ature and time of cooling. 

Again, the number of students in a given institution from year 
to year depends on too many causes for the relation to be ex- 
pressed by an equation. But the curve can be plotted and thus a 
clear and condensed representation of the variation can be secured. 
The attendance at Johns Hopkins University from 1877 to 1904 is 
given in the subjoined table. 



1877 89 


1884 249 


1891 468 


1898 641 


1878 104 


1885 290 


1892 547 


1899 649 


1879 123 


1886 314 


1893 551 


1900 645 


1880 159 


1887 378 


1894 522 


1901 651 


1881 176 


1888 420 


1895 589 


1902 694 


1882 175 


1889 394 


1896 596 


1903 695 


1883 204 


1890 404 


1897 520 


1904 715 



A glance at the corresponding curve (Fig. 8) gives however a 
much clearer picture of the growth of the institution. 



Analytic Geometry. 



25 



PROBLEMS. 

1. The following table gives the number of years one may expect 
to live at the ages indicated : 



Age 


Expectation 


Age 


Expectation 


Age 


Expectation 





39.9 


35 


29.4 


70 


8.4 


5 


49.1 


4i» 


26.0 


75 


6.4 


10 


47.0 


45 


22.7 


80 


4.9 


15 


43.1 


50 


19.5 


85 


3.7 


21) 


39.4 


55 


16.4 


90 


2.8 


25 


3G.1 


60 


13.5 


95 


2.1 


30 


32.7 


65 


10.8 


100 


1.6 



Plot the corresponding curve. 

2. The temperature of a fever patient was as follows : 



Julv 8, 


5 


P. M. 


99.4 


July 


13, 


12.30 


P. M. 


106.3 


Julv 9, 


6 


A. M. 


98.0 


July 


13, 


1.30 


P. M. 


105.6 


Julv 9. 


5 


P. M. 


105.0 


July 


13, 


5 


P. M. 


104.6 


July 10, 


6 


A. M. 


99.0 


July 


14, 


6 


A. M. 


98.2 


July 10, 


5 


P. M. 


99.2 


July 


14, 


5 


P. M. 


98.6 


July 11, 


6 


A. M. 


98.2 


July 


15, 


6 


A. M. 


98.0 


July 11, 


1 


P. M. 


106.0 


July 


15. 


5 


P. M. 


99.0 


July 11, 


2 


P. M. 


104.0 


July 


16, 


6 


A. M. 


98.0 


Julv 11, 


5 


P. M. 


103.6 


July 


16, 


5 


P. M. 


98.4 


July 12, 


6 


A. M. 


98.0 


July 


17, 


6 


A. M. 


98.0 


July 12. 


5 


P. M. 


98.4 


July 


IT, 


5 


P. M. 


98.6 


July 13, 


6 


A. M. 


99.0 


July 


18, 


6 


A. M. 


98.0 



Plot the corresponding curve. (In this problem it will evi- 
dently be wise to take as the axis of X the line corresponding to 
some high temperature, such as the normal temperature of 98 
degrees, in place of the line corresponding to zero degrees.) 

4. Select some prominent stock or article of produce and plot 
its prices for the next two weeks as given in the daily stock jr 
market reports, giving the reasons for any important fluctuations. 

16 - In these examples there is no particular 

choice OF scales reason why the same scale should be used 
OF measurement. n both axes. The fact that a certain dis- 
tance has been used to denote a year or a 
day is no reason why the same distance should be used 
to denote a dollar or a degree of temperature. Mathe- 
maticians are consequently accustomed to select such scales 
as are most convenient, and to indicate their choice bv a foot 



26 Analytic Geometry. 

note as in Fig. 6, or by figures on the axes as in Figs. 7 and 8, 
The choice is usually so made as to make the important features 
of the curve prominent. Thus in problem 2 the physician will 
choose a small distance to represent the day and a much greater* 
one to represent the degree ; while the curve of problem 1 may be 
almost wholly deprived of interest by the choice of a large dis- 
tance to represent the unit of age and a small one to represent 
the unit of expectation. It is sometimes necessary after a curve 
has been drawn to one scale to make magnified drawings of certain 
portions in order to examine more closely certain doubtful points. 

17 - When both the assumptions made in para- 

determination graph 13 can be granted, pairs of values 
OF intermediate intermediate between those actually known 
values. can De determined with a close degree of 

approximation by measuring the co-ordi- 
nates of the corresponding point on the curve. Thus in problem 1 
of paragraph 15 the expectation for any intermediate age can be 
determined by measuring the ordinate corresponding to the ab- 
scissa representing that age. But in many cases this method of 
procedure is not in order since the assumptions of paragraph 13 
cannot be made. In the Johns Hopkins problem for example the 
attendance does not pass continuously from 520 to 641, taking 
all the intermediate values, but passes by leaps from one value 
to another. In other words the actual locus is not a continuous 
line as we have drawn it, but a succession of disconnected points. 
The second problem of paragraph 15 is a case in which the 
second assumption of paragraph 13 cannot be granted. While 
it is true that the temperature varies continuously with the 
time, it is by no means to be expected that our curve, based upon 
observations made at intervals of several hours, shows all these 
variations. There may be and probably are intermediate vari- 
ations of which we have no record. In such a case as this where 
the nature of the phenomena is not sufficiently well understood 
to enable us to deny the existence of these intermediate vari- 
ations, as we might in the age-expectation problem, and where 
the laws regulating the phenomena do not admit of algebraic 
expression, there is of course no way of determining the number or 
location of such intermediate variations. When the locus is 
determined by an equation they may be determined by the aid 
of the differential calculus. 



Analytic Geometry. 27 

18 - The question of continuity, raised inei- 

definition AND dentally in the last paragraph, demands 
test of careful consideration. In order that a curve 

continuity. ma y be continuous it is necessary that as x 

(the abscissa of the tracing point) varies 
continuously, y (the ordinate of the tracing point) shall also vary 
continuously; or, expressed algebraically, y is a continuous func- 
tion of x when the change in y due to a change in x may be made 
as small as we please by taking the change in x small enough. At 
all points where this condition is satisfied y is said to be a con- 
tinuous function of x, at any point where it is not satisfied, y is 
said to be discontinuous. The fever temperature problem affords 
us an example of a continuous function. If the change in the time 
be small enough the change in the patient's temperature will be as 
small as Ave please, while in the Johns Hopkins problem no 
shortening of the interval will make the difference between two 
successive values of y a small quantity. 
Let us examine the continuity of a simple function,* say 

If x increases by an amount Ar, or as we more frequently 
say, takes an increment Ar, y takes an increment which we 
may call Ay. 
Then y — 2a? 2 

V + Ay = 2(a? ■+ AS) 2 = 2z +. ±xNx + 2 A® 

therefore Ay == 4a? Ace + 2 Ax • 

So long as x remains finite, the right hand side of this 
equation tends to zero as Ax tends to zero, or in other words 
Ay may, by a proper choice of A#, be made as small as Ave 
please for all finite values of x. That is, y is a continuous 
function of x so long as x is finite. 



*If the student is not familiar with the ideas and notation of mathe- 
matical functionality, he should at this time read Appendix B. 

tThe method here used is applicable to the most complex forms, but 
numerous algebraic difficulties are encountered in the attempt to employ it. 
The overcoming of these difficulties falls in the province of the differential 
calculus. 



28 Analytic Geometry. 

19 - The definition of continuity given above 

infinite brings to light also another sort of discon- 

discontinuity. tinuity. If the student will plot the curve 

he will note that as x tends to unity, y passes beyond all 
limit (i. e. ; becomes infinite), but so long as x differs 
ever so little from unity y remains finite. That is, no matter how 
small the increment which carries x from its previous value to 
the value unity, y leaps from a finite to an infinite value. The 
function is accordingly said to have an infinite discontinuity, 
while the discontinuities previously discussed are called finite 
discontinuities. Such functions as the student will meet in the 
present work are, as may be shown by the calculus, free from finite 
discontinuities, and such infinite discontinuities as may occur 
can be detected by plotting the curve.* 

PROBLEMS. 

Find the values of x for which the folloAving functions have 
infinite discontinuities. 

w 9 

i. y 



x — 3 



2. y 



(x — l){x — 2) 
3. y = sec x 



*The Johns Hopkins problem and problem 4 of paragraph 15 furnish 
examples of finite discontinuities, but in all these cases the functional re- 
lation is not given, so that they constitute no exception to the statement 
made above. As an example of a function with a finite discontinuity con- 
sider the equation 

1+ex 

As x (regarded as positive) tends to zero, y tends to zero; but as x (re- 
garded as negative) tends to zero, y tends to unity. The corresponding locus 
has therefore a finite jump from unity to zero as it crosses the Y axis. 



CHAPTER VI. 

TO DEDUCE THE EQUATION WHEN THE RESTRICTIONS 
ON THE MOVEMENT OF THE TRACING POINT ARE 

GIVEN. 

20 - Problems of this type by no means always 

the general occur in the simple form indicated in the 

method. heading of this chapter. Very frequently 

we are confronted Avith a mere verbal de- 
scription of a finished curve, containing apparently no refer- 
ence to the method of its construction. But if the ver- 
bal description is complete it contains all the limitations 
on the movement of the tracing point. The method of meeting 
the problem is therefore always the same. Consider the curve as 
traced by a variable point; determine the law which regulates 
the movement of the point, (i. e., the condition which is satisfied 
by all points on the curve and by no others) ; state this condition 
in algebraic form, i. e., express it as a relation between the 
variable co-ordinates, x, y, of the variable point. We have then 
an equation of condition between x and y which is satisfied by the 
co-ordinates of all points on the curve and by no others ; in other 
words we have the equation of the given curve. 

The student may get a somewhat clearer idea of this process if 
he will consider the analogy between it and the work of trans- 
lating from one language into another. In translation from 
English into German for example, the stu'dent must have not only 
a knowledge of the German words equivalent to the English words 
in the passage to be translated, he must also have a knowledge 
of the peculiar forms, the idiomatic constructions, of the two 
languages. Ordinarily he will first of all throw the English 
sentence into the German order and replace the English idiom by 
the corresponding German idiom, and then is ready for the actual 
work of translation. Now algebra is after all to a great degree 
merely a language, and an equation is a sentence. The equation 
of a locus is the statement in algebraic language of the conditions 



30 



Analytic Geometry. 



under which the tracing point moves, and the deduction of such 
an equation is merely the translation of the ordinary English 
description of those conditions into algebraic language. For 
example consider the circle of radius 2 centered at the point 
(4, 3). The algebraic idiom requires first of all that the curve 
be described as the locus of a moving point, and we accordingly 
throw our description of the curve into the new form, "A variable 
point moves in such a way as to keep its distance from the point 
(4, 3) equal to 2." The phrase, "A variable point" translates at 
once into, "The point (x, y)", and if we knew an algebraic equiv- 
alent for "The distance from {x, y ) to (4, 3)" we should at once 
equate it to 2 and have the equation of the locus. 

Since the definitions of the simpler curves are largely stated 
in terms of distance and direction, it will be wise before we take 
up the work proper of this chapter to develop some fundamental 
formulae which will enable us to translate questions of distance 
and direction into algebraic language. 



21. . 
DISTANCE 
BETWEEN TWO 
POINTS IN TERMS 
OF THEIR CO- 
ORDINATES. 



Consider any two points P t and P 2 whose 
co-ordinates are (x x , y x ) and (x 2 , y 2 ). Draw 
P t P 2 . Draw P ± Q ± and P 2 Q 2 parallel to OY 
and P t R parallel to OX. Let D be the re- 
quired distance. Then we have by direct 
application of our geometry 



But 
and 

therefore 



B = P 1 P 

■L \-tv ==z ^fi^/2 ' == ~ 2 *^1 

=y-i — yi 



v P,R +P 2 K 



P 2 R 



D=^(x 2 —x 1 ) 2 +{y 2 — y 1 ) 



22. 

CO-ORDINATES OF 
A POINT WITH A 
GIVEN DISTANCE 
RATIO. 



Given two 
points P x and 
P 2 , to find the 
co-ordinates of 




■X\ 



Fig. 9. 



a third point P 3 on the straight line joining 
P ± and P 2 which shall have with respect to P x and P 2 a given dis- 
tance ratio. (See paragraph 9.) 



Analytic Geometry. 



31 



Let the co-ordinates of P u P,, I\, 
be (.' M //,\ (x 2) //J, (-/';,, //.,) and let the 



numerical value of 



p.p. 



be 



The distance ratio P 3 with respect 
to /\ and P., will then be either 

— ^ or -^ according as the point P 3 

K a 2 

is within or without the segment 

Consider the case where P 3 is 
within the segment P X P 2 . We have 



"T 



V 



Fig. 10. 



at once 



p,p, 
p,p. 






therefore 



A = <*— "gi 



and 






a^ 2 + A^ 
\ + a 2 



PROBLEMS. 



1. Show from the same figure that i/ 3 = -^ — 2, ' x . 

A l + A 2 

2. Construct the figure for the case when P 3 is without the 
segment P X P 2 , and show that the same formulae hold. 

3. In the work of both paragraphs 21 and 22 the axes have been 
rectangular. Deduce the corresponding formulae when the axes 
are oblique. 

4. Find the distance from (2, 3) to (5, —1) ; (4, 1) ; (—5, 1) ; 
(-1,-1); (0,3). 

5. What is the general formula for the distance of a point from 
the origin ? 

6. Express the co-ordinates of the middle point of a segment of 
a line in terms of the co-ordinates of its extremities. 

7. Find the co-ordinates of the points which divide the segment 
of the line terminating at (1, 3) and (4, — 2) into three equal parts 
and find the length of these parts. 



32 Analytic Geometry. 

8. Find the co-ordinates of the points which have the given dis- 
tance ratios with respect to the following pairs of points. The 
shorter segment is in each case the one terminating at the point 
first named. 

Points. Ratios. 



2 

2 

(2,-1) (-2, -4) 2 

(5,6) (1,-3) -| 

__7 

9 



(1, 4) (2, 3) 
(3, 2) (-1, 0) 



(0, 0) (—2, 5) 



23. In analytic geometry the angle which a line 

angle between makes with the X axis is always measured 
A line and the f r0 ni the positive end of the X axis toward 
X AXIS. the positive end of the Y axis. Remembering 

this, the student should have no difficulty in 
showing that the tangent of the angle made with the X axis by the 
line through two points is given by the formula, 



tan 



J/o — t/i 



where is the angle and (x 19 y- ± ) 9 (x 2 , y 2 ) are the points. He 
should also show that this formula holds for all possible positions 
of the line. 

PROBLEM. 

Find the angles which the lines of problem 8 of the last para- 
graph make with the X axis. 

24. Now that we have developed our formulae 

ILLUSTRATIONS OF f or distance, distance ratio, and direction of 
the development a straight line we are ready to take up again 
OF equations OF the problem we were compelled to leave un- 
LOCI. finished in the latter part of paragraph 20. 

We now are able to translate the phrase, "The distance from (a?, y) 
to (4, 3)" by the expression V (x — 4) 2 -\-(y — 3) 2 , and the state- 
ment of the way in which the variable point traces the curve now 
translates into 

V(# — 4) 2 + (t/ — 3~p = 2 



Analytic Geometry. 33 

which is the equation of the curve, since it is the algebraic state- 
ment of the necessary and sufficient condition that the point 
(x, y) may lie on the circle. 

It is usual to reduce such expressions however to the simplest 
possible form, and the equation of this circle would usually be 
written in the form 

# 2 + V 2 — 4# — 2y — 11 = 0.* 

Let us consider a second example of this sort. The line joining 

a variable point to the point (1, 2) makes with the X axis an angle 

whose tangent at any instant is equal to the abscissa of the variable 

point at the same instant. Find the locus of the variable point. 

"The variable point" translates into (x, y). "The abscissa of the 

variable point" translates into x. "The tangent of the angle made 

by the line with the X axis" translates (by paragraph 23) into 

y — 2 

-. Therefore the algebraic statement of the condition under 

x — 1 & 

y 2 

which the curve is described is evidently - = x 

i.e., y = x 2 — x-\-2. 

If the student desires to know the form of the curve it can easily be 
plotted. Later on he will learn to classify simple curves without 
plotting. , 

PROBLEMS. 

1. Find the locus of all points equally distant from (1,1) and 
(2, 4) ; from (1, 3) and (—1, 5). 

2. Generalize problem 1 by taking (x t , y ± ) and (x 2 , y 2 ) as the 
two fixed points, and show that the equation is always of the first 
degree. 

3. Find the equation of the circle whose center is at (a, b) and 
whose radius is r. (Since proper choice of a, ~b, and r will make 
this any circle whatever, the corresponding equation is called the 
general equation of the circle.) 

4. Generalize the second illustration of this paragraph by replac- 
ing the point (1, 2) by a point {x ly y x ). 

5. A point moves so that the square of its distance from (3, 2) 
plus the square of its distance from (1, 3) equals 27, find the 
equation of the locus. Does a comparison of the result with that of 
problem 3 give any hint as to the nature of the curve ? 

6. Show that if a point moves so that the sum of the squares of 
its distances from three fixed points is constant, the equation of 



^See appendix C. 



34 Analytic Geometry. 

its path will always be of the second degree, will haye no term in xy } 
and will have the same coefficient for the terms in x 2 and y 2 . Can 
these statements be extended to a greater number of points? 

7. Q 1 and Q 2 are two fixed points and P is a variable point. The 
movement of P is subject to the condition that the tangents of the 
angles which the two lines PQ X and PQ 2 make with the X axis shall 
be numerically equal but of opposite sign. Find the locus of P. 

8. A moving point traces a straight line passing through the 
point (1, — 2) and making with the X axis an angle whose tangent 
is 2. Show that the equation of the line is 

y = 2x — 4 

9. A line has an intercept on the Y axis of 4 (i. e., passes through 
the point (0., 4)) and makes with the X axis an angle whose 
tangent is 3. Show that its equation is 

y = Sco + A. 

10. Since any line may be defined by its intercept on the Y axis 
and the angle it makes with the X axis, generalize problem 9 and 
show that the equation of any straight line is of the first degree 
and of the form 

y = mx + h. 
What are m and h? 

11. Show conversely that any equation of the first degree in x 
and y can be reduced to the form 

y = mx + h 

and therefore represents a straight line. 

The work of this paragraph will be resumed after the student has 
acquired a greater amount of material on which to base problems. 



CHAPTER VII. 

TO DEDUCE THE EQUATION WHEN A FINITE NUMBER 
OF POINTS ON A LOCUS OF SOME KNOWN TYPE IS 

GIVEN. 

25 - If we are given merely a number of points 

limitations OF on a locus there is no way by which the 
the problem. equation may be deduced, but if in addition 

outline OF THE to a number of points we are given such ad- 
method. ditional information as will enable us to 

determine the general form of the equation, the problem is at once 
simplified. For if the form of the equation is known, each point 
that satisfies the equation gives us a relation connecting the 
coefficients; and the complete determination of the equation is 
therefore possible whenever a sufficient number of points have been 
given. For example, let it be known that a straight line passes 
through (4, 7) and (3, 5). Problem 10 of the last paragraph tells 
us that the equation must be of the first degree and therefore of the 
general form 

Ax+By+C = 0* 

Now if the two points lie on the locus their co-ordinates must 
satisfy the equation and we have 

4|+7|+1=0 

3|+of+l=0 

4 R 

whence — = — 2 and — '== 1 

and the equation of the line is 

— 2x + y + l = 0. 



*This form contains apparently three arbitrary constants, but division 
by any one of them reduces the equation to a form which contains only two. 
Such an equation is said to contain two effective constants. 



36 Analytic Geometry. 

Again, problem 3 of the last paragraph shows us that the equa- 
tion of the circle whose center is at (a, b) and whose radius is r 
is of the form 

(x — a) 2 +{y — b) 2 = r 2 . 

This equation contains three effective constants, but it is of the 
second degree in these constants, and our subsequent work will 
therefore gain in simplicity if we reduce the equation to the form 

x 2 + y 2 — 2ax — 2ly + a 2 + b 2 — r 2 = 0, 
or, putting a 2 -f- b 2 — r 2 = — c, 

x 2 -f- y 2 — 2ax — 2by — c = 0. 

If now the three points (1, 1), (3, 3), (4, 1) lie on the curve we 
have 

2a + 2b + c = 2 
6a + 6& + c = 18 
8a + 2b + c = 17 

whence the student may find a, b, c and so determine the equa- 
tion of the circle. 

26 - The fact that the co-ordinates of every 

number OF POINTS" point on a curve must satisfy the equation 
required TO f the curve, taken in connection with the 

determine A algebraic theorem that n non-homogeneous 

CURVE. equations are necessary and sufficient to de- 

termine n unknown quantities, leads us at once to the important 
theorem : 

The number of effective arbitrary constants in an equation is 
equal to the number of arbitrary points through which the corre- 
sponding curve may be made to pass. 

PROBLEMS. 

1. Find the equations of the straight lines through the follow- 
ing pairs of points : 

(1,3) (2.-1); (2,4) (3,0); (4, 3) (2, 4) ; (4, 3) (-4,-3). 

A 7? 

In the last case the values of —and — are infinite. This of 

C C 

course means merely that C is zero. The difficulty may be avoided 
by dividing by A in place of C. 

2. Find the equation of the circle through the points (1, 2), 
(2,4), (1,4). 

3. A curve of the form y 2 = 2px passes through the point 
(4, 2) . Determine the value of p. 



Analytic Geometry. >H 

4. Show that the equation of any straight line through the 
origin is of the form y = mx where m is an arbitrary constant. 
What is the geometric significance of m? 

5. Find the equation of the circle through the three points 
(-1,2), (-1,-3), (0,0). 

6. Show that the equation of any curve which passes through 
the origin can have no constant term. 

.7. How many effective constants are there in each of the follow- 
ing equations? 

ax 2 + by 2 + 2hxy + 2gx + 2f y + c = 
(aa> + i)y + c)(da; + fy + g)=0 
a(bx + cy + d)=0 

{a J r b)x -{- cy = 
ax 2 -\-~by — ex -j- 2 = 

27 '■ In place of giving points on the curve, some 

comple. other condition may be stated which is equiv- 

CONDITIONS . ' , . . J . x -r^ 

equivalent TO a l en t to giving one or more points. For ex- 
two OR more ample, to give the center of a circle is equiv- 

simple ones. alent to giving both a and b, and therefore 

is equivalent to giving two points on the 
curve. Again, to give the angle which a line makes with the 
X axis is to determine m, and therefore is equivalent to giving 
one point. 

PROBLEMS. 

1. Find the equation of the circle which passes through (3, — 1) 
and has its center at (4, 2) . 

2. Find the equation of the circle which passes through (3, — 1) 
and (1, 4) and has its center on the X axis. 

3. Find the equation of the straight line through the point 
(3, 4) , making an angle with the X axis of 70°, 110°, 45°, 135°. 

4. A given line makes with the X axis an angle whose tangent 
is m, and has an intercept a on the X axis. Show that its equation 
is 

y = m(x — a). 

5. Find the equation of a line through (1, 5), making an angle 
of 45° with the X axis. 

6. Find the equations of the lines through ( — 2, 5) parallel to 
the lines of problem 3. 

7. Find the equation of the line through the point ( — 3, — 4) 
parallel to the X axis, to the Y axis. 



CHAPTER VIII. 



TRANSFORMATION OF CO-ORDINATES. 



28. 
THE GENERAL 
PROBLEM. 



of the following : 



If the student turns back to paragraph 10, 
he will note that our present system of co- 
ordinates rests on certain assumptions which 
are equivalent to the arbitrary determination 
the origin, the direction of one axis, 
the angle between the axes, the scales of measurement. 
The exigencies of the discussion may at any time demand a change 
in any one of these, and the algebraic significance of such a 
change must therefore be investigated. As in the case of our 
work in one dimension, we desire to find formulae which will give 
us the old variables in terms of the new. We consider in turn 
the formulae corresponding to changes in each of the four as- 
sumptions mentioned above. 

29. To change the first assumption without 

movement OF the producing any change in any of the others 
axes parallel TO it is sufficient to move the axes parallel to 
themselves. themselves. Let XOY be the original system 

and by such a movement secure a second 
system X'O'Y', where the co-ordinates 
of 0' referred to XOY are a and &. 
Let P be any point in the plane and 
let the co-ordinates of P be (x,y) in 
the first system and {x\ y) in the 
second. Then for all positions of P 

we have 

x = x + a 

y=y' + ~t> 

which are accordingly the formulae 
of transformation. 



■> 


r i 


( 


? 












,0 






■0 




Fl 


s. 11. 



XI 



*The subdivision of our subject adopted in paragraph 12 seems to call at 
this point for a discussion of the last of the problems there stated, but the 
discussion is in many cases so facilitated by transforming the axes that it 
seems wise to introduce the present chapter at this point. The student who 
is not thoroughly familiar with the simpler theorems concerning the pro- 
jection of plane contours should read Appendix D before undertaking the 
work of this chapter. 



Analytic Geometry. 39 

PROBLEMS. 

1. Kind the formulae corresponding to a change to a new 
set of axes parallel to the old, but with the new origin at the 

point Cl, - D, (4, 0), (0, 4), (- 2,-8), (T, 1). 

2. Find the formulae of transformation corresponding to the 
following movements of the axes : 



i & 



The X axis 1 upward, Y axis 4 to the right; 
The X axis 4 downward, Y axis unmoved ; 
The X axis unmoved, Y axis % to the left. 

3. What movements of the axes correspond to the following 
substitutions? * 

< x = x + 4 \x = x — 2 

x == oo -f- 6 c # = a?' 

y = y' — 2 \y = y' + 4: 

4. In the figure the axes are rectangular. Will the same form- 
ulae hold in case the axes are oblique? 

5. By a movement of the X axis reduce the equation 

y = 3x-\-4 : 

to an equation in x and «/' which has no constant term. (Assume 
the axis to be moved a distance a, make the substitution, and then 
determine a by equating the constant term in the transformed 
equation to zero.) 

6. Do the same thing by a movement of the Y axis. 

7. Free the equation 

cc 2 + y 2 — 2x — 5y + l = 

from the terms of the first degree in x and y by a movement of the 
origin, keeping the axes parallel to their original position. 

8. Move the origin so that the equation 

l y z + 2fy + 2gx + c = 

shall be transformed to an equation having no term, of the first 
degree in y and no constant term. Is such a movement always 
possible? 

9. What must be the co-ordinates of the new origin in order 
that the most general equation of the second degree 

ax 2 +ly 2 +2hxy + 2gx + 2fy + c==0 



40 



Analytic Geometry. 



may reduce to an equation which has no terms of the first degree 
in the variables?* Is the operation possible when 

h- = al)f 

10. Are there any terms of an equation which cannot be removed 
by a transformation of the kind we have been considering? 



30. 



To change the second assumption without 



ROTATION OF THE affecting any of the others it is sufficient to 
axes. rotate the axes about the origin into a new 

position in which they make an angle with 
the old position. 

Let XOY be the axes in the first 
position and X'O'Y' the axes in 
the new position. Let P be any 
point having the co-ordinates (x, *{ 
y) and (%', y) in the two systems. 
Then 

Oa =®, P. a =y. 
On = x\ Pc = y . 

The two contours OaP and OcP 

have the same terminal points and \ 

their projections on any line are therefore equal. Hence we have, 

by projecting on OX, 




Oa cos + aP cos 



Oc cos 0-\-cP cos (0 



)• 



2 2 

Similarly by projecting the same contours on the line OY we have 



aP cos 0= Oc cos (^ — 0)+ cP cos 0, 



Oa cos - 

and these two equations at once reduce to 

x = x' cos — y sin 
y = x' sin + ij cos 
which are the desired formulae of transformation. 



*If an equation of the second degree has no terms of the first degree in 
the variables it is of the form 

ax^ -f Irjz 4_ 2hxy + c = 0. 
If such an equation is satisfied by a point {% , y ) it will also be satisfied 
by the point ( — x , — y ). The locus is therefore symmetrical with respect 
to the origin, and our problem might be thus stated : To move the origin 
to the center of symmetry of the curve represented by the equation of the 
second degree. 



Analytic Geometry. 41 

If we desire the formulae which correspond to a change 
from X'OY' to XOY we may soive these equations for x and 
y\ or we may project on the lines OX' and OY', or we may note 
that this new case differs from the original one only in the angle, 
which is now negative. Any one of these methods will give us 

x' = x cos + y sin 
y' = — x sin + y cos Q 

PROBLEMS. 

1. Write the formulae of transformation which correspond to 
the following rotations, putting in in each case the numerical 
values of cos and sin 

60°, 30°, 45°, —30°, 150°, (tt — 60°), (f^— 30°Y ^ , «■'. 

2. Show that the equation 

x- — xy — 2=0 

may be freed from the term in xy by a proper rotation of the axes. 
(Apply the proper formulae and the equation becomes 

x' 2 (cos 2 + cos sin 0) + y' 2 { sin" — cos sin 0) 

+ xy (cos 2 — sin 2 — 2 sin cos 0) — 2 = 0. 

In order that there may be no term in xy we must so choose 
that 

cos 2 — sin 2 — 2 sin cos = 
or cos 20 — sin 20 = 

tan 20 = 1 

= 22° 30' == - . 
8 

3. Free the equation 

x 2 — xy -\-Sy — 2x = 

from the term in y by a rotation of the axes. 

4. How many terms may be removed from an equation by a rota- 
tion of the axes? Are there any terms which are unaffected by 
such a rotation? 

5. Through what angles may the axes be turned without intro- 
ducing an xy term into the equation 

cc 2_ y 2 ==r 2? 

6. Show that the general equation of the second degree will 
reduce to an equation without an xy term if the axes are rotated 

through an angle such that tan 20 = " . 

O — b) 



42 



Analytic Geometry. 



31. 



CHANGE OF THE 
ANGLE BETWEEN 
THE AXES. 



To change the third assumption without 
affecting any of the others it is sufficient to 
change the direction of either of the axes, 
but it will be better to develop the more gen- 
eral formulae corresponding to a change of 
the directions of both axes. 

Let XOY be the first set of axes 
and X'OY' the second. Let OX' make 
an angle a and OY' an angle fi with 
OX, and let the angle between OX' 
and OF be S ( S= — a). Let P 
be any point {x, y) in the plane. 
Then by considering the projections 
of OaPtmdi OcP on OX and OTwe 
have 

x = x cos a + y cos fi 

y = x sin a -\- y sin fi, fig. 13. 

a set of formulae which are sufficient to transform from any set 

of rectangular axes to any set of oblique axes which have the same 

origin. 

If we wish to transform from oblique to rectangular axes we 

have only to solve the formulae just derived for x and y and we 

have 

/ . x sin fi — ■// cos fi 

x . ~ 

Sm o 

i . — x sin cl -\-y cos a 




sin S 



1. Free the equation 



PROBLEMS. 



2#-^3i/ + l=0 



from its term in y by a change in the direction of the Y axis. 
2. What form does the equation 

cd 2 -f- 7/ 2 = r 2 

take when the X axis is left unchanged but the Y axis moved so 
that the angle between the axes is 30°? 



Analytic Geometry. 43 

32. To change the last of our assumptions 
change OF THE without affecting any of the others it is 
scales of sufficient to write 

MEASUREMENT. 

x = Kx 
y = Ly'. 

The effect is evidently to multiply the unit of measurement on 
the X axis by K and on the Y axis by L. 

33. The transformation from any system of 
the general axes to any other may now be accomplished 
transformation, by the proper combination of the formulae 

developed in the preceding paragraphs. 

PKOBLEMS. 

1. Write the formulae of transformation which correspond to 
a movement of the origin to the point (4, 5) and a rotation of 
the axes through 60°. 

2. Write the formulae which correspond to the transformation 
from rectangular axes to a new set of oblique axes, whose origin 
is at ( — 4, 3) and wiiose X and Y axes make angles of 30° and 95° 
with the original X axis. 

3. The Origin is moved to the point ( — 2, — 4), and the axes 
are rotated through an angle of 10°. W>ite the formulae of 
transformation. 

4. Show that none of the transformations so far discussed can 
change the degree of an equation. (It is sufficient to show that it 
cannot be raised. For if a change of the axes transforms an 
equation into one of lower degree, change the axes back to their 
original position and the equation will be restored to its original 
form, i. e., the degree of the transformed equation will be raised. 
Therefore when the student has shown that the degree of an 
equation cannot be raised he has shown that it cannot be lowered.) 

34. Each of the above transformations has 
TRANSFORMATIONS been interpreted as corresponding to a 
interpreted AS change in the system of co-ordinates. There 
changes OF THE is however another interpretation which is 
LOCI. frequently adopted. Consider any equation 
as referred to a given system of reference. Apply any one of the 
above substitutions. The result will be a new equation, which 
may of course be referred to the original system of reference and 



44 Analytic Geometry. 

when so referred represents a new locus. The transformation 
may thus be regarded as a change in the locus instead of a change 
in the system of reference. For example, apply to the circle 

(» — a)«+(y — 6}« = r« 

the substitution 

x = x -\- a y = y -f- & 

and refer the resulting equation 

x'~ -\-y' 2 =r 2 

to the original set of axes. From this point of view the trans- 
formation has evidently resulted in moving the center of the circle 
from the point (a, b) to the origin. The student will find it in- 
teresting to study all of the above transformations from this 
second point of view. 



CHAPTER IX. 

INTERSECTION OF LOCI. 

35 - We have found that any equation connect- 

THE significance i n g x an <j y represents a locus every point of 
OF I magi N ARIES. which satisfies the equation. The following 
♦ question, which was given a somewhat super- 

ficial treatment in paragraph 11, now demands more careful 
consideration. "Given two equations, is it possible to find 
a point or points which lie on both loci and therefore 
satisfy both equations? 1 ' Looked at from the geometric side 
the question is, "Do the loci intersect?-' and the answer is, "They 
may or may not according to their relative positions." Looked at 
from the algebraic side the question is, "Can two equations in two 
variables be simultaneously satisfied?" and the answer is, "Yes, 
without exception." The cause of this apparent discrepancy lies 
in the nature of our fundamental assumptions, which were so 
made as to establish a one to one correspondence between the 
points of the plane and pairs of real values of x and y, while the 
theorem that two equations in two variables can always be simul- 
taneously satisfied holds true only when pairs of imaginary values 
are included. 

To make the matter a little clearer consider the equation 

x — y = 0. 
Pairs of values that will satisfy this equation are of two kinds; 
real values such as x = a, y = a } or* complex values such as 
x== a -\- ib, y = a -\- ib, including pure imaginaries x = ib, y = ib. 
The first kind alone corresponds to points in the plane and includes 
all the points on the line through the origin bisecting the angles 
in the first and third quadrants. It is evident that there is 
not a complete correspondence between this line and the equation 
x — y=0, since the equation has. a much more general significance 
than the line; and it is also evident that this lack of complete 
correspondence is due to the nature of our fundamental assump- 
tions, which give us no geometric representation for pairs of 
imaginary values. 



46 



Analytic Geometry. 



It is however frequently desirable to be able to state algebraic 
theorems in geometric language, and so mathematicians are ac- 
customed to speak of these pairs of complex values of x and y as 
represent ing imaginary points. From this point of view the curve 
corresponding to any equation /(./-. y)=0 is considered to consist 
of : (a) an infinity of real points which constitute the visible curve : 
(1)) an infinity of imaginary points just as intimately associated 
with the equation, but having no representation in the diagram. 
(See appendix E.) 

Any point, real or imaginary, which belongs to each of two 
curves is called an intersection, real or imaginary, of the curves. 
From the algebraic theorem that two equations of degree in and n 
in two variables can always be satisfied by mn pairs of values of 
the variables it follows that two curves of degree m and n intersect 
in mn real or imaginary points, some of which may in special 
cases coincide with each other. 



36. 
POSSIBILITY OF 
ERROR. 

and 



So long as both the equations are of the 
first degree no ambiguity in the results is 
possible. For example, the two equations 



'l.T 

±x 



3 = 
7 = 



yield on solution the two equations 

x = 2 y = l 
showing that the intersection is on the line parallel to the Y axis 
at a distance 2 to the right, and on the line parallel to the X axis 
at a distance 1 above. 

But consider the intersections of Y 

the circle of Fig. 14, Avhose equation 
is 

2x 2 + 2y 2 = 3 

and the curve A'B'C'D' whose equa- 
tion is 

x- + xy + if = 2. 

Solving these two equations for x 
we have 



\ 



6±!/ 2 




8 



Fig. 14. 



Analytic Geometry. 47 

showing thai the points of* intersection arc on one of the four* 
lines .1.1'. BB\ CC\ l)D'. If now 



V 



6-1 20 



8 ' 

for example, be substituted for x in the equation of the 
circle wo obtain, on solving the resulting equation in y, they 
co-ordinate of the intersection .1, but we also obtain the 
y co-ordinate of the point A" in which we have no interest; 
while if the same value of x is substituted in the equation 
of the other curve, we obtain the y co-ordinates of the points 
A and A' . In such cases as this the student must determine 
by substitution in the equations which of the points obtained 
correspond to the actual intersections of the curves. 

PROBLEMS. 

Find the intersections of the following pairs of curves : 

1. 2x + Sy + 1 ■ = 2. (B — y = 

4:00 — y + 2 = x + y = 

3. x —2y + 1 = 4. x — iy + 2 ■ = 

— 4^ + 2// — 7 = m + iy+A = Q. 

5. x — 2iy + 3 = 6. x 2 + y 2 = 4 

w + 2iy + 3 = x + 2y = 

7. a? 2 + i/ 2 + 4 = 8. x 2 + i/ 2 — 2 = 

a> + 2y -h 1 = ^4-^ — 4 = 

9. 4,z 2 + %y 2 — 12 = 10. 4^ 2 + 9?/ 2 = 36 

3,r 2 — 4// 2 — 12 = 9# 2 + 4// 2 = 36 

11. Find the intersections of 

y = a 
with x 2 -\-y 2 = l) 2 

and state in hoth algebraic and geometric terms what happens 
as a, at first less than 1), gradually increases till it is greater than 
6. 

12. Find the intersection of 

y = m x x -f- &i 
and y = m 2 x -j- & 2 

ard discuss the case when m^ = m 2 . 

Find the intersections of the following curves with the axes : 

13. y = mx + 6 14. — + -^ = 1 

15. Aa? + B// + C = 16. a? 2 — 2ay-f-l = 



CHAPTER X. 

THE EQUATION OF THE FIRST DEGREE AND THE 
STRAIGHT LINE.* 

37 - Our investigation of the geometric prop- 

standard forms erties and relations of loci by means of their 
OF the equation equations will be facilitated by adopting 
OF THE straight some mode of classification of equations. 
line. That by degrees is probably the most natural 

and for our present purpose the most convenient. 

The student has already deduced for himself a number of im- 
portant results concerning the equation of the first degree and its 
corresponding locus which are here summarized for convenience 
of reference. 

I. Every equation of the first degree represents a straight line. 
(Problem 11. Paragraph 24.) 

II. Conversely, every straight line is represented by an equa- 
tion of the first degree. (Problem 10. Paragraph 24.) 

III. If the equation be of the form 

y = mx -f- h 

m is the tangent of the angle made by the line with the X axis 
and h is the intercept on the Y axis. (Problem 10. Paragraph 
24.) 

IV. If the equation be of the form 

y = m(% — a), 

m has the same meaning as before, but a is now the intercept of 
the line on the X axis. (Problem 4. Paragraph 27.) 



*We begin at this point the study of the last of the five problems men- 
tioned in paragraph 12 and shall continue it for several chapters. The 
student must remember, however, that any such division of a subject as is 
attempted in stating these five heads is of necessity somewhat arbitrary, and 
he must therefore not be surprised to find under this last head problems 
and theorems that might with, entire propriety be stated under some other 
heading. In particular he will find an entire chapter (Chapter XII) devoted 
to loci problems which have been deferred to this later position because the 
student had not at an earlier date sufficient material on which to base them. 



Analytic Geometry. 49 

Y. If the equation be of the form 

»+*-=i 

a o 
a and b are the intercepts on the X and Y axes. (Problem 14. 
Paragraph 3G.) 

The tangent of the angle made by the line with the X axis is 
frequently called the slope of the line, and the forms in III and IV 
are consequently called the slope equations of the straight line, 
while the form in V is called the intercept equation of the straight 
line. Ax + By + C = is called the general equation of the 
straight line. 

PKOBLEMS. 

1. Find the slopes and intercepts of the following lines : 



2x + 3t/ + 1 = 
x — 2y — 4 = 


y = 4a? -f- 2 

x = 3ij + A 
y = Z 


^ — 2^ = 


x = a 



2. Compare the general equation with the slope and intercept 
equations and deduce the folloAving results for the general equation 

Slope = — ^ 

C 
Intercept on the X axis = — 

P A 

C 
Intercept on the Y axis = — 

3. A knowledge of the intercepts makes the plotting of the line 
an easy matter, excepting in one special case. (What is this ex- 
ception?) Plot the lines of problem 1. 

4. Since the slope of a line depends on the ratio — what con- 

h 
elusion may be drawn as to the equations of parallel lines ? 

5. Let the lines AB and CD have the equations 

y = m r T -f- 7l t 

and y = m 2 x -f- h 2 

respectively. Show that 

tan0- m '- 



1 + ???! m 2 
where is the angle between the lines. 



50 Analytic Geometry. 

6. Find the angles between the following pairs of lines : 

(D + 2y — 1 = -l.r — 2y + 1 = 

ff — % + ^ = x = 3y + 2 

±%+y — 2 = i\r — 2// + 1 = 

2/ = 3a? — 4 a? — 6/y + 4 = 

a? + y — 2 = # — 2y + 3 = 

4a? + 3 = — Ay y + 2a? + 5 = 

7. Show that the angle between the two lines 

A 1 w + B^+O i = Q 
and A 2 x + # 2 £/ + ^ Y 2 — 

is given bv 

+ /, B 1 A l —B t A, 

tan ^tcT^;t 

8. From the results of problem 5 determine what conditions 
must be satisfied by the coefficients of 

y = m x x + h^ 
and y = m 2 x + h 2 

in order that the lines may be parallel. Perpendicular. 

9. Determine the corresponding conditions for 



and 




A 2 x + B 2 y + C 2 = 


= 
= 


10. 


Remembering that 








tan 







sm u — 

± 1 


and cos 




x 1 + tan 2 6 





±1/ 1 + tan 2 

deduce formulae for the sine and cosine of the angle which a 
line makes with the X axis in terms of m and h. In terms of A 
and B. 

11. A line is subject to the condition that it must be parallel 
to the line 

2x — 3j/ + l = 0. 

To what extent are its coefficients determined and to what extent 
are they still arbitrary? 

12. Find the equations of lines through the point (2, 3) parallel 
and perpendicular to each of the following lines : 

w — 2y + 3 = x — 2.?/ + 3 = 

# = 3?/ + 4 a? = 

— 2/+2a?=9 co~ y = 

x = 4y ax — cy + f = 



Analytic Geometry. 51 

L3. Arc there any cases of parallelism or perpendicularity 

among the following lines? 

x — 2y + 3 = x + 4 y + 12 = 

^ = 4a? + 5 6 y — '6x + 1 = 

lx — 2y + 3 = yr-4:X +3 = 

14. A line is subject to the condition of passing through the 
point (1, 3). To what extent are its coefficients still unde- 
termined? 

15. Find by the method of paragraph 25 the general equation 
of a straight line through the point (x lf y 1 ). 

38 - If we attempt to find the equation of the 

THE equation OF straight line through the two points (x lt y,) 
THE straight LINE and {x 2 , y 2 ) , the method of paragraph 25 is 
through TWO of course perfectly rigorous, but it has the 

given points. disadvantage of demanding an amount of 

algebraic work which is somewhat wearisome, and in certain sim- 
ilar but somewhat more complex investigations becomes well nigh 
prohibitive. It is sometimes possible in such cases to avoid much 
of this algebraic work and infer the form of the equation desired 
from principles already established. Consider for example prob- 
lem 15 of the last paragraph. The equation desired must be of the 
first degree, must be satisfied by the point (j> y\), and must be 
sufficiently general in its form so that we may be able to satisfy 
it by the co-ordinates of any other one point in the plane. The 
equation must of course not be identically zero. Any equation, 
no matter how secured, that possesses these properties is the 
equation desired. Xow it is easy enough to manufacture an 
equation which possesses them. 

y — y 1 = m(x — x 1 ) 

in fact possesses all these properties and will hereafter be used 
as the equation of any line through (x x , y x ). (What is the geo- 
metric significance of m?) 

The problem stated at the heading of this paragraph may be 
treated in the same way. Here the equation must be of the first 
degree, must be satisfied by (x 1} y x ) and (x 2 , y 2 ) 7 must contain no 
arbitrary constant (Why?), and must not vanish identically. 

(V — yi)(®2 — #i)=(0?-— xj(y 2 — y x ) 

(V — Vo) (x 1 — x 2 ) = (x — x 2 )(y 1 — y 2 ) 



52 Analytic Geometry. 

meet these conditions, and either of them may therefore be taken 
as the desired equation. The same result may be obtained by 
substituting in 

y — y x = in{x — x x ) 

the value of m deduced in paragraph 23. These equations are 
more frequently written in one or the other of the two forms 



!/ — 


Vi 


— Ih. 

x,. 




^(x 

Xy 


y — 


- //i 


_ •'' 




Xy 



7/2 i/y X-2 Xy 

Still another form of this equation is frequently met in mathe- 
matical literature. It may be deduced as follows: If the point 
(x,y) is to trace a straight line it must satisfy an equation of the 
form 

ax -\- by -\- c = 0, 

and if the line is to pass through the points (x ± , y x ) and {x 2 , y 2 ) 
we must have 

ax x -\- by x -j- c = 
and ax 2 + by 2 + c = 

The co-existence of these three equations is the necessary and suffi- 
cient condition that the point (x, y) may trace a straight line 
through the two points (x t , y x ) and (x 2 , y 2 ). But the necessary 
and sufficient condition for the co-existence of these three equa- 
tions is 

x y 1 

®i y± 1 =0 

x 2 y 2 L 
which is therefore the equation of the line. 

39. Consider a line through (x ly y-^) making 

THE straight line the angles a and 13 with the axes, and let 
given BY two ( Xj y) be a variable point on the line. The 

equations IN necessary and sufficient conditions which 

three variables, must be satisfied in order that the variable 
point may be on the line are 

x — x 1 = r cos a 
and y — y±=r cos (3 



Analytic Geometry. 



53 



where r is the variable distance from (a?, y) to (x ly y x ). These two 
equations may be written in the form 



;c — ,i\ y_ 

cos a 



Vi 



or 



cos/? 

x = x x -f- r cos a 

t/ = 2/i+r cos p 
These two equations are used in cases where the student is inter- 
ested in. the distance from the tracing point to some fixed point 
on the line, r can be readily eliminated between the two equations 
and any one of the forms of the equation previously discussed at 
once deduced. Hereafter we shall refer to this form as the para- 
metric form of the equations of the straight line. 

40 - The problem of finding the distance from 

:R0M A a given point (x 1} y x ) to a given line 



POINT TO A LINE. 



Ax + By + C = 



might now be solved by writing the equation of a line through 
( x u Vi) perpendicular to 

Ax + By + C=0 7 

finding the point of intersection of these two lines, and determin- 
ing the distance from this point of intersection to the point 
{x 1} y t ). But a geometric solution is simpler and leads to an 
important algebraic result. 

Let P be any point (x lf y ± ), and AB any line. Let p' denote the 
perpendicular distance from the point P to the line AB, and p the 
perpendicular distance from the origin to the same line. Let a 
and P denote the angles which the perpendicular from the origin 

Y Y 





Fig. 15. 



Fig. 16. 



on the line AB makes with the axes. Draw Pa perpendicular to 
OX, Pl> and Od perpendicular to AB, and etc parallel to AB. 



54 Analytic Geometry. 

Then Oc + bP — Od = Pe 

or Oe+ bP — Od = — Pe 

i. e. ? j- t cos a + ^/i sin a — P = V or — p' 

according as the point P is on the opposite or same side of the line 
AB as the origin. This formula gives the distance from the point 
to the line in terms of the co-ordinates of the point and the 
constants p and a Avhich determine the position of the line. 

41 ■ The necessary and sufficient condition 

NORMAL FORM OF that any point (x, y) be on the line, (i. e. 

the equation OF the equation of the line) is that p' for thai 

A straight line, point shall be zero. 

x cos a -\- y sin a — p = 

is therefore another form of the equation of the straight line. It 
is called the normal form of the equation. 

In order to reduce the general equation to the normal form we 
multiply it by an undetermined constant m. Then if 

mAx + mBy + mC=0 
is identical with 

x cos a -j- y sin a — p = 
we must have 

1)1 A = COS a 

mB = sin a 
m C= z — p 
Therefore m 2 A 2 + m 2 B 2 = 1 

and . m = — . 



1 A 2 + jB 2 



Substituting this value we have for the normal form of the equa- 
tion 

Ax By C _ 



i A 2 + B 1 i A' + B 2 i A 2 + B 2 

The results of the last two articles may evidently be stated in 
the following condensed form. The distance of any point from a 
given straight line is the value obtained by substituting the co- 
ordinates of the point in the left hand member of the normal 
form of the equation of the line. 



Analytic Geometry. 



PROBLEMS. 



1. Write the equations of each of the following lines in each 
of the forms already developed. 



Through the point 
(1- 3) 



Angle ivitli X axis. 



:>> 



A) 



3 radians 

65 degrees 

c degrees 



(1, 4) 
(—2, 
(a, b) 

Through the Points 

(1, 4) and (2,-3) 

(1, —6) O, £) 

(4, c) (0, 2) 

(4,5) (1,-3) 

(4, 2) (— a, 2) 

Through the origin parallel to the first of the list. 
Through (1,1) perpendicular to the first of the list. 

2. Find the distance from the origin to each of the lines in the 
first group above. 

3. Find the distance from the point (1, 1) to each of the lines 
of the first group above. In which cases are the origin and 
(1, 1) on opposite sides of the line? 

4. Find the area of the triangle whose vertices are (0, 0), 
(1,2), (3,1). 

5. Find the area of the triangle whose vertices are (1, — 1), 
(3,2), (-3,-2). 

6. If A is the area of a triangle and (w ± , y x ), (x 2 , y 2 ), (# 3 , y 3 ) 
are its vertices, show that 



2A = y 1 (x 2 —x 3 )+y 2 (x s — x 1 )-^y 3 (x 1 —x 2 ) 
7. In determinant notation this becomes 



x x 


y± 


i 


x 2 


y-2 


i 


x 3 


Vz 


i 



2A = 



Deduce this form directly by use of the determinant form of the 
equation of a straight line. 

8. To say that three points lie on a straight line is evidently 
equivalent to saying that the area of the triangle formed by them 
is zero. Show that the necessary and sufficient condition for 
collinearity of three points (w ± , y x ), (x 2 , y 2 ), (a? 3? y 3 ) is 

1 



x 2 y 2 1 

#3 Vz 1 



= 0. 



dG Analytic Geometry. 

9. Deduce the same conclusion from the fact that the necessary 
and sufficient condition that the three points may lie on the line 

Ax + By + C = 

is that the co-ordinates of each point shall satisfy the equation of 
the line. 

42 PKOBLEMS. 

INTERSECTIONS OF „. , , . „ , „ „ 

L|NES i Find the intersections of the following 

lines : 

1. 3x — 2y + 2 = and x + y=2 

2. x — Zy + \ = and x — ±y — 1 = 

3. x = 2y — 7 and y = lx — 2 

4. Find the vertices of the triangle formed by the lines 

y = ±x 

2x — Sy = 4 

5x — 4^+2 = 

5. Show that the necessary and sufficient condition which must 
be satisfied in order that the three lines 

A 1 x+B 1 y + G 1 = 
A 2 x+B 2 y + G 2 = 

A 3 x+B 3 y + C 3 = 

may meet in one point (i. e., be concurrent) is 

= 



A t 


B 1 


C 


A, 


B, 


C, 


A s 


B, 


c, 



In general any two lines intersect in one point. The only case 
in which an apparent exception is to be noted is when the lines 
are parallel. It is possible to include this apparent exception, 
however, under the general case. If the student will refer to his 
discussion of Problem 12, Paragraph 36, and take into consider- 
ation the geometric significance of m 1 and m 2 he will see that the 
result of that discussion may be stated thus. As two lines tend 
to parallelism one or both of the co-ordinates of the point of 
intersection must increase beyond all limit, i. e., the point of inter- 
section removes indefinitely from the origin. Mathematicians are 
accustomed to use in place of this statement the abbreviated 
phrase, "Parallel lines intersect at infinity" and in this way are 
able to include the special case of parallelism in the general 
statement that anv two lines intersect in a single point. 



Analytic Geometry. 57 

43. Let the straight lines represented by the 

FAMILIES OF LINES, two equations 

A 1 x + B 1 y-\-C\ = 
and A 2 x + B 2 y + C 2 = 

intersect in the point (a? 15 y x ). Then if // and v are any two 
constants, 

p{A x x + B l2 / + (7J+ v (A 2 x + £ 2 t, + C 2 )=0 

is the equation of a straight line (Why?) passing through the 
point uv //J. since the substitution of x x and y t for <r and y 
causes each of the parentheses to vanish and therefore satisfies 
the equation. If we put 



A x x + B lV + C t 


= 8 1 


A 2 i 


v + B,y 4- C 2 


= S 2 



we may write the third line in the abridged form 

. S 1 + \8 2 = 0. 
In general if R ± = and I\. z = are the equations of any two loci. 

R l -f \R 2 = Q 

is the equation of a third locus passing through all the inter- 
sections of the first and second. 
The equation 

#i+ \s 2 = o 

contains one arbitrary constant as it properly should, since the 
line represented by it has been subjected to the single condition 
of passing through one point and therefore has one degree of 
freedom remaining. This constant may be determined so that the 
line passes through any other point in the plane, and therefore 
it is evident that the equation by proper choice of A may be 
made to represent any line in the plane through the intersection 
of $!=() and & 2 = 0. The aggregate of all such lines is spoken 
of as the family of lines through the intersection of 8 1 =Q and 
S 2 = and the equation 

is called the equation of the family. The arbitrary constant is 



58 Analytic Geometry. 

called a parameter, a name given to any constant entering into 
an equal ion and taking in the course of the discussion a succession 
of arbitrary values. Thus 

y = mw 

y = a 

y — y x = n(x~x 1 ) 
are respectively the equations of the families of lines which pass 
through the origin, are parallel to the X axis, and pass through 
the point (i\< y ± ). The respective parameters are m, a, and n. 
If it is desired to determine a particular member of a family 
an additional condition must be given. When the equation of 
the family is subjected to this condition a value or set of values 
of the parameter will be determined which Avill determine the 
member or members of the family satisfying the condition. For 
example, suppose we wish to determine that member of the family 

A ± x + B ± y + C x + X(A 2 x + B 2 ij + C 2 )= 
which passes through the point (3, 2). We have at once 

3A X + 2B 1 + C x + X(SA 2 + 2B 2 + C 2 ) = 
whence the value of X is at once determined. Substituting this 
value of X in the equation of the family we have the particular 
member of the family which we desire. 

PROBLEMS. 

1. Write the equation of the family of lines through the inter- 
section of 

2% — 3y + 1 = 
and 5a? — y + 2 = 

and find that member of the family which 

(a) passes through the point (1, 5) 

(b) is perpendicular to 4a? — 2y'=0' 

(c) has an intercept of 4 on the X' axis 

(d) makes an angle of 30 degrees with the X axis. 

(e) is parallel to the bisector of the 1st and 3rd quadrants at 
the origin. 

2. Write the family of lines through the point (1, 2). 

3. Write the family of lines parallel to the line 

3oj + 2^ + 4 = 0. 

44. If the equation of a line be written in the 

the line AT intercept form 

INFINITY. y__±_ JL =Z -(_ 

a ■ b 
and the line be removed farther and farther from the origin, a 






Analytic Geometry. 59 

and b indefinitely increase and the coefficients of x and y lend to 
zero. In other words, as a line removes indefinitely from the origin 
ils equation tends to the form 

0x + Oy — 1 

or, since the equation may be multiplied by any constant what- 
ever, to the form 

0x + 0y=C. 

Moreover this form does not depend on the original position of 
11k 1 line or on the manner in which it is removed. In other words 
the equations of all lines tend to a single form as. the lines are 
indefinitely removed from the origin, a statement which is equiv- 
alent to the assertion that all points at infinity satisfy the single 
first degree equation 

Ox + Of/ = C. 

But since in analytic geometry we deal with loci only through 
their equations, mathematicians are accustomed to express all 
this by the abbreviated phrase, "All points at infinity lie on a 
single straight line whose equation is 

For any point in the finite part of the plane the left hand member 
of the above equation is of course zero, and since we are in 
almost all of our investigations concerned only with the finite 
part of the plane, it is customary in all such investigations to 
abbreviate still more the above form and write the equation of the 
line at infinity as 

(7 = 0. 

45. The subject of parallel lines affords an 

parallel lines. interesting application of this idea of the 
line at infinity. Let the line GH be drawn 
through the intersection of AB and ^ 
CD,. Then if the equations of AB and 
CD are respectively 

S 1 = and & 2 = 
the equation of GH is 

S 1 + XS 2 = 0. FIG. 17. 

Let CD be removed indefinitely, its equation will tend to (7 = 0. 




60 Analytic Geometry. 

GH will tend to parallelism with AB, and its equation will tend 
to the form 

fi[ 1 +.X(7=0, 

which gives our former theorem that the equation of a line parallel 
to a given line differs from the equation of the given line only in 
the constant term. A line parallel to a given line is, from this 
new point of view, a line through the intersection of the given line 
with the line at infinity. 



CHAPTER XI. 

THE CIRCLE, A SPECIAL CASE OF THE EQUATION OF 
THE SECOND DEGREE. 

46. The most general form of the equation of 

the general the secon( j degree is 

EQUATION OF THE 

circle. ax* + by 2 ^2Kxy + 2gx + 2fy + c = 

and the corresponding curves, from certain relations which they 
bear to the cone, are called conic sections or simply conies. 

Before taking up the study of the general conic we shall con- 
sider the special case of the circle. Its equation has been already 
found (Problem 3, Paragraph 21), in the form 

(x — a) 2 -\-(y — b) 2 = r 2 
where (a, ~b) is the center and r the radius. Multiplying, trans- 
posing, and replacing the constant term by c, we reduce this 
equation to the form 

# 2 + V 2 — 2«# — 2by + c = 
or multiplying by an arbitrary constant 

Ax 2 + Ay 2 + 2Gx + 2Fy + = 0. 

We therefore state the general theorem that every circle is repre- 
sented by a second degree equation, without a term in xy, and 
with the same coefficient for the terms in x 2 and y 2 . The student 
may deduce the converse theorem that every second degree equa- 
tion without a term in xy and with the same coefficients for the 
terms in x 2 and y 2 represents a circle by showing that every such 
equation may be reduced to the form 

(% — a)*-+(y — l) 2 = r 2 . 

PROBLEMS. 

1. Compare the two forms 

Ax 2 + Ay 2 + 2Gx + 2Fy + G = 
and (x — a) 2 + (y — ~b) 2 = r 2 

and hence show that the center is the point ( , — — ) and 

A A 



the radius is -±- \/G 2 + F 2 — AG. 
A 



62 Analytic Geometry. 

2. Find the centers and radii of the following circles : 

#2 _|_ y2 _ ± x _|_ 5y _ 12 = 

3a? 2 + 3?/ 2 — 2t/ = 

a? 2 + 2/ 2 — 4a? = 

4a? 2 + 4v/ 2 — 12a? + 2y + 4 = 

3a? 2 + 3if — 4a? + 2 = 

x ~ + i/ 2 + jP# + ( ii/ — s = o 

3. Find the equations of the circles with the following centers 
and radii. 

Centers • Radii 

(3, 4) 2 

(4, 1) | 

(3,0) 5 

(-2,-1) k 

(0,-4) sin-^ 

o 

4. Write equations which by proper choice of the constants in- 
volved will represent any circle of radius 2 whose center is on the 
X axis, on the line y = a, on the line y = 3a?, on the line y = 3a? + 2, 
on the circle a? 2 -f- y 2 = 4. 

5. Write the equation of the< circle centered on the curve y 2 = 3a? 
and passing through the points (2, 3) and (4, 4). 

6. Apply to the circle 

a transformation of co-ordinates which will make both the new 
axes tangent to the circle. What is the new form of the equation? 

7. What is the form of the equation when the new axes are a 
diameter and the tangent at its extremity ? 

8. Write the equation of a circle centered at (3, 1) and tangent 
to the line 

3a? — 2t/ + 4 = 

(See paragraph 41.) 

9. Show by a geometric construction that the condition 

(a? — a) 2 + (y— h) 2 = r 2 

is satisfied by the co-ordinates of all points on the circle of radius 
r centered at («, Z>) and by no others. 

47. We might find the intersections of two 

intersections OF circles by a direct solution of the two equa- 
circles. tions for a? and y, but this would introduce 

an unpleasant amount of algebraic work. Subtracting^ 

a? 2 + y 2 + 2a x x + 2b x y + c, = 
from a? 2 -j- y 2 + 2a 2 x + 2b 2 y + c 2 = 

we have 2x(a 2 — a 1 ) + 2y(b 2 — h 1 ) + c 2 — c 1 = 0, 



Analytic Geometry. 63 

a new locus passing through the intersections of the two circles. 
(See Paragraph 43.) This locus is a straight line and must 
therefore be the common chord of the two circles. Our problem 
is now reduced to the simpler one, already" solved, of finding the 
intersections of this line and either of the circles.* 

When one of the circles lies wholly within or without the other 
the intersections are of course imaginary. It will probably sur- 
prise the student to find that the common chord is, however, 
always real. The explanation of the fact will be evident if he will 
find the equation of the line determined by a pair of conjugate 
imaginary points, e. g. {a + ib , c -f- id) and (a — ib, c — id). 

PEOBLEMS. 

' Find the intersections of the following pairs of circles : 

1. x 2 +y 2 + 2x — 3*/+2 = 
(x — 3) 2 +(2/ — 4) 2 = 16 

2. ( x — l)*+(y— 2) 2 = 4 
x 2 -f- y 2 = 4 

3. (x — iy + (y— 2) 2 = 4 

(# + 3) 2 +(?/ + 5) 2 = l 

4. (x— 6) 2 +(2/ — 4) 2 = 4 
(x-2) 2 +(y-l) 2 = 9 

48 - While the definition sometimes given in 

tangents and elementary geometry of a tangent line to a 

normals. circle as a perpendicular to a radius at its 

extremity is entirely correct and might be used as a basis 
for our discussion of the tangent, it is not a definition 
which admits of extension to other curves which we shall 
study. The tangent line to any curve may evidently be 
regarded as the limiting position of a secant line as two of 
the points of intersection of the curve and the secant tend 
to coincidence. Let a secant line meet a curve in two points P 
and Q. and let the point P tend to coincide with Q. The limiting 



*Two circles have of course four points of intersection (paragraph 35), 
but two of them are always imaginary points at infinity. The subtraction 
above gave terms Oa? 2 and Oy 2 . To drop these as we did was to assume that 
x and y were to remain finite, and the resulting equation therefore is satis- 
fied by the finite intersection of the circles, but not by the infinite ones. 



64 Analytic Geometry. 

position of the secant line as P tends to Q is the tangent at Q. 
This is sometimes expressed by saying that the tangent to a curve 
meets it in two coincident points. 

The normal to a curve at any point is the perpendicular to the 
tangent at that point. 

Among the questions that arise concerning tangents are two 
that decidedly outrank the others in importance. 

1. Given the equations of a line and a circle, how shall we 
determine whether the line is tangent to the circle, or in other 
words, what is the condition which the coefficients of the line 
must satisfy in order that it may be a tangent line to the circle?. 

2. Given the co-ordinates of a point and the equation of a 
circle, what is the equation of a line through the point tangent 
to the circle ? 

49. Any satisfactory definition of the tangent 

condition OF will of C01irS e lead to the condition of tan- 

gency if properly considered. For example, 
the fact that the tangent is perpendicular to the radius at its 
extremity is equivalent to the statement that the perpendicular 
distance from the point (a, h) to the line 

y = mx + h 

must equal r if the line is tangent to the circle of radius r centered 
at (a, 1)). Applying this test we have as the condition of tangency 

ma — 1) -\- h 

\/l + m z ~ V 



or v h = ± r V 1 + m ' z — ma + & 

This method of deriving the condition of tangency is unfortu- 
nately applicable only to the circle, since the definition of tangency 
on which it is based does not hold for other curves. The definition 
of the tangent as a line meeting the curve in two coincident points 
holds however for all curves. In deducing the condition of tan- 
gency from this definition we find the intersections of the line and 
the curve. The co-ordinates of these intersections are given by 
ordinary algebraic equations in one variable and the necessary 
and sufficient condition for tangency is that these equations shall 
have equal roots. Thus 

y = mx + h 
meets x 2 -f- y 2 =r 2 



Analytic Geometry. $5 

in two points whose x co-ordinates are given by the quadratic 

equa t ion x l -j- m 2 x 2 + 2mwh + ^ 2 — r 2 = 0. 

The necessary and sufficient condition for coincidence of the 
points of intersection, and therefore for tangency of the line, is 
that this quadratic in a? be a perfect square, i. e., that 

»V — (h 2 — r 2 ) (1 + m 2 ) = 
or ft = :±ryi+ m 5 

as before. 

50 - Given a point (a? 1? t/J and a circle 

EQUATION OF THE 2 g 

TANGENTTHROUGH ^ ~"~ ^ ? ' 

A given point. to determine the equation of the tangent to 

the circle through the given point we first 
write the general equation of a line through the point 

y—y 1 = m{x — x 1 ) 
or y = mx -\- y x — mx v 

If this line is tangent we must have from the last paragraph 



y x — mx x = ± r\/l -\- m* 



x 1 y 1 ± r ya? 1 2 +y 1 2 — r s 



i. e., m— 22 

x 2 — r 2 

Inserting these two values of m in the equation of the line we 
have the equations of the two tangents from (x x , y x ) to the circle. 
If however (se 1} y x ) is on the circle, we have 

and therefore m = ^ 

Vi 

Substituting this value of m in the equation of the line Ave have 

for the equation of the tangent at a point (x ly y x ) on the circle 

x 2 -\- y 2 =r 2 

the form y — y{ = — (x — x x ) 

i. e., yy 1 J rX x 1 = r 2 

This method of finding the equation of the tangent is theoretic- 
ally general, but in the case of more complex equations we en- 
counter serious algebraic difficulties. For the development of the 
equation of the tangent at a point on a curve a second method, 
based upon the definition of the tangent as the limiting position 



66 



Analytic Geometry. 



of the secant, is worth our investigation. Consider then a point 
P, (#u 2/J < on the circle, and give to x x and y 1 such increments 
Aa?i and A//i, that the new point 
Q, (a? x + Aa? x , ft + Aft), so ob- 
tained shall also be on the circle. 
Then the line 






*i) 




Fig. 18. 



is the secant line through the two 
points (x 19 ft) and (a? x + A^, 
ft + Aft). If we can determine 

the limiting value of the ratio =^ 

Ax, 
as Aa?i and consequently Aft 
tend to zero we shall have the 
slope and therefore the equation 

of the tangent at (a? x , y x ). Since both points are on the circle we 
have 

*i a + &'='' 

and x x - + 2a?! A», + Aa?, 2 + ft 2 + 2ft Aft + Aft 2 = r 
Subtracting the first of these equations from the second we have 





S^A.-n + Aa?! + 2ftAft + Aft" 



whence 



Aft 
Aa?! 



2a?, + A. 



2ft + Aft 

But the limit of this fraction as /\x x tends to zero is — -^ which 

is therefore the limiting value of the slope of the secant line 
through the two points as the second point tends to coincidence 
with the first, i. e., the slope of the tangent at (x x . ft). The equa- 
tion of the tangent is therefore 

y— i/i=— — (* — »i) 
ft 
reducing as before to 

yy 1 -\-xx 1 = r 2 

This method is that employed in the differential calculus, and 
by the aid of the processes elaborated in the discussion of that 
subject is applicable to the more complex forms which present 
too much algebraic difficulty for our former method. 



Analytic Geometry. 



67 



51. The distance from the point of tangency 

sub-tangent and to the point where the tangent intersects the 
sub-normal. x axis is called the length of the tangent, 

and the projection of this portion of the tangent line on 
the A axis is called the sub-tangent. The distance from 
the point of tangency to the point where the normal in- 
tersects the X axis is called the length of the normal, 
and the projection of this portion of the normal line on 
the X axis is called the sub-normal. Thus if the center of the 
circle be at the origin, PT, PO, OQ and QT are respectively the 
lengths of the tangent, normal, sub-normal and sub-tangent at P. 
Let the radius of the circle be r; the co-ordinates of P be x x , y x ; 
and of T be ao 2 , 0. Then in this special 
case it is evident that the lengths of 
the normal and subnormal are r and 
x x . To find the lengths of the tan 
gent and sub-tangent we write the 
equation of the tangent at P and find 
its intersection with the X axis. The 
length of the sub-tangent is then 



x x 



and the length of the tangent is 

V(*2 — XiY + yJ 




j 



*r + tt' 



■Mi 
x, 



Fig. 19. 



r\ 



results which might have been directly obtained by the aid of trigo- 
nometry. 

PROBLEMS. 



1. What is the condition of tangency to the circle when the 
equation of the line is given in the form 

Ax + By-\-C = Q? 

2. Find the equations of the tangents to 

through (1,3), (5, 6), (1,-1). ' 



68 Analytic Geometry. 

3. Find the tangents to 

(x-l)^(y-2)* = d 

through the point (5, 3). 

4. {Show from the quadratic for determining m that the tangents 
through P are real or imaginary according as P is outside or 
inside the circle. ^ 

5. Show that the equation of the tangent to „ 

x- + 2/ 2 .+ 2aa? + 2fy/ + c = 
at the point (x 1 , y t ) is 

y — y,=~ *±\i* — m x ). 

This form reduces to 

«NPi + VVi + ax + by = x x 2 + y ± 2 + ax 1 4- by t 
Adding ax\ + by x + c to both sides, the second member van- 
ishes (why?) and we have a frequently used form 

6. Show that the equation of the normal to the circle 

a?2 _L y2 = r 2 

at the point (x 19 y x ) is 

Note that the normal to the circle always passes through the 
center. 

52 - Given any two tangents to a circle, the 

POLES and polars chord joining their points of tangency is 
defined. called the chord of contact. We have just* 

seen that any point determines two tangents to a circle 
and hence it determines a chord of contact. Similarly 
any chord of a circle determines the two tangents at \ts 
points of intersection with the circle, and hence it de- 
termines a point, the intersection of the two tangents. In other 
words there exists a one to one correspondence between the chords 
of any circle and the points of the plane, so that to each point there 
corresponds a single chord and conversely. The chord is called 
the polar line or merely the polar of the point with respect to the 
circle, and the point is called the pole of the line with respect to the 
circle. 



Analytic Geometry. 



69 



53. The equation of the polar of the point 

equation of the (a^ y x ) w ith respect to the circle 

FOLAR " «»-tV = H 

might be derived cfirectly by finding the points of contact of the 
t tangents from (x 1} y x ) and writing the equation of the line through 
these two points; but the algebraic work is somewhat complicated 
and we will accordingly make use of a method similar to that of 
paragraph 38. 

Let P be any point (x ± , y x ) and 
let the two tangents from P to the 

circle 

x 2 -f- y 2 = r 2 

touch the circle at the points A and 
B, {x 2 , y 2 ) and (a? 3J y 3 ). Since PA 
is tangent -at (a? 2 , y 2 ) its equation is 

Similarly the equation of PB is 

* ®® 3 +yy 3 =^r 2 . 

But each of these lines passes 
through (x ly y x ) and therefore we FlG- 2 o. 

have 

x 1 x 2 -\-y 1 y 2 = r 2 

®i® 3 + ViVz = r 2 - 
The problem before us is to find an equation of the first degree 
in x and y which is satisfied when x and y are replaced either by 
a? 2 and y 2 or by x 5 and i/ 3 . An inspection of the pair of equations 
last written shows that 

xx 1 + yy 1 = r 2 

is such an equation. It is therefore the equation of the polar 
of the point {x x , y x ) with respect to the circle 

x 2 -f- y 2 = r 2 . 

When the point (af t , y x ) is on the circle the equation of the polar 
becomes the equation of the tangent at {x ly y ± ). In other words 
,the tangent is only a special case of the polar, being the polar of 
the point of tangency. By allowing the point P in the figure to 
approach the circle the student can convince himself that the 
tangent is the limiting position of the polar as the pole approaches 
the circle. 




70 



Analytic Geometry. 



The discussion given above in no way depends on the location 
of the point P outside the circle. If P is inside the circle the two 
tangents are imaginary and the points of contact also imaginary; 
but if the pole P and the circle are real the points of contact are 
conjugate imaginary points and the polar is real, a fact which 
is also evident from the equation. 

54 - To find the pole of a given line with respect 

CO-ORDINATES OF to the circle 
THE POLE. a? 2 + t/ 2 = r 2 

we may also use a method shorter than the direct one of finding 
the intersection of the two tangents having the given line as chord 
of contact. Let the given line be 

ax -j- by + c = 
and assume the co-ordinates of its pole to be (w x , y x ). Then the 
equation of the line must be 

s»i + Mi = I* 2 - 
Since these two equations represent the same line their co- 
efficients must be proportional (why not equal?) i. e., 

®i = Mi = ZZJ^ 
a o c 



whence 



ar 



Vi 



hr 



55. 
POLAR AS LOCUS 
OF HARMONIC 
CONJUGATES. 



c c 

Consider any point P and draw through P 
a line meeting the circle in the points Q and 
R. Let S be the harmonic conjugate of P 
with respect to Q and B. The locus of S as 
the line rotates through P is the polar of P with respect to the 
circle. 

The proof of this theorem assumes the following : 

(1) Problem 7, paragraph 9 ; 

(2) The roots of ex 2 -\- bx -\- a = i) 
are the reciprocals of the roots of 
ax 2 -f- b x -f c — ; , 

(3) The sum of the roots of ax 2 + bx 

+ c = is — -^-. 
a 

Let the co-ordinates of P be (x t , y x ) 
and the equation of the circle be fig. 21. 

x 2 -\- y 2 =r 2 




Analytic Geometry. 71 

Write the equation of the line through P in the form 

x = x 1 -\- p cos a 

y = Vi + p cos ft 

Substitute these values of x and y in the equation of the circle, 
and the distances PQ and PR from the point P to the intersections 
of the line and the circle are given by the quadratic 

P 1 (cos" a -f cos 2 ft) -\- 2/o (a cos a -f g/, cos ft) + -r/ 2 + y? — r = 
-, 1 , 1 = 20?', cos a + ?A cos ft) 

PQ' 1 ' PR x'+yf — r* 

Let the co-ordinates'of S be (a?', y") then 



The necessary and sufficient condition that S shall be the harmonic 
conjugate of P with respect to (,) and 11, i. e., the equation of the 
locus of S, is therefore 

— 2 (#! cos a + j/ 1 cos ft ) 2 



ft ~~ r i/U'i — ^) 2 + C^i— y) 



But cosa = 



cos ft 



j/ (y — a», ) 2 + d/ — y x f 



II 



i (af — x x Y + {y' — y x Y 
Substituting these values Ave have for the equation of the locus 

i. e., ^4-2/^^r 2 

as before. 

56 « An important problem here presents itself. 

polar as LOCUS Given a fixed point and a circle, to find the 
OF poles. locus of the poles with respect to the given 

circle of all lines through the given point. This locus is evidently 
a definite curve and therefore its equation must be a single re- 
lation between the co-ordinates of the variable point and known 
constants, but without arbitrary parameters. Let the point be 
(#!, y x ) and the circle be given in the form 

x 2 -\-y 2 = r 2 . 

The equation of any line through the point is 

y—y l = m(x — x 1 ) 



12 Analytic Geometry. 

and the co-ordinates of the pole are 

— nir 1 > r 2 

x = // ~ 



mXj + t/ l — in.i\ + //, 

We have here two equations connecting the co-ordinates of the 
variable point with known constants and with the arbitrary para- 
meter m. What we desire is a single relation free from arbitrary 
parameters, connecting the co-ordinates with each other and whii 
known constants. We therefore eliminate m and find for the equa- 
tion of the desired locus (after dropping accents) 

showing that the locus of the poles of all lines through the point 
(% x , v/j is the polar of (o? 1? y x ). 

PKOBLEMS. 

1. Find the polars of the following points with respect to 

$~ + y 2 = r2 
(1,3), (—2,4), (k,—p), (0,a), (6,sinfc), (c, ^) . 

2. Find the poles of the following lines with respect to 

x 2 + if = 10. 

Zx — 2?/ + 4 = »_4.^ = i 

3.x — 7*7 + 4 = a l~ 

y = mx -\-l) x — y = 

\y === 5 + 6r 

3. Find the general equation of the polar of a point on the J 
axis with respect to the circle 

a> 2 + 2/ 2 = >' 2 ; 

of a point on the line x = 2\ on the line x = 3y, on the line 
x = Zy + 2. 

4. Given two diameters at right angles to each other show th«t 
the polars of all points on one are parallel to the other. R* 1 *! 
conversely that the poles of all lines parallel to one lie on the 
other. 

5. Show that if the point (x 1} y x ) lies on its own polar with 
respect to 

x- -f- y 2 = r 2 
it lies also on the circle. 



Analytic Geometry. 73 

(>. Show that the condition which must be satisfied in order 
that (w 1} 7/J may lie on the polar of (x 2 , y 2 ) is identical with the 
condition which must be satisfied in order that (x 2) y 2 ) may lie 
on the polar of (x ly yj, and thus prove the following theorem. 

Let A and B be two points. Then if A lies on the polar of B, B 
lies on the polar of A. 

7. On the basis of the theorem just stated deduce a method of 
constructing with ruler and compass the pole of a given line 
which does not meet the circle in real points. 

S. Construct with ruler and compass the polar of a point inside 
the circle. 

9. Show both geometricly and algebraicly that the polar of the 
center of the circle is the line at infinity. 



CHAPTER XI. 

ADDITIONAL WORK ON THE SUBJECT OF LOCI. 

57 - Problems in which the restrictions on the 

GENERAL REMARKS movement of the tracing point are given and 
ON LOCI problems, the equation of the locus demanded are ail 
alike in the fact that the method of solution consists 
merely in the translation of the law of movement of 
the tracing point into algebraic language. They may, how- 
ever, be divided into two general classes. In the first 
class fall problems of the type discussed in paragraph 24, in which 
the statement of the law gives the locus immediately. In the 
second class fall problems of the type discussed in paragraphs 
55 and 56, in which the attempt to translate the law of movement 
of the point leads to relations connecting the co-ordinates of the 
tracing point with each other and with certain arbitrary para- 
meters. 

In every legitimate locus problem in plane geometry the number 
of equations expressing such relations, either between the vari- 
able co-ordinates and the parameters, or between the parameters 
themselves, is always one more than the number of the parameters, 
so that it is possible by the elimination of the parameters to 
deduce a single relation connecting the co-ordinates of the tracing 
point with each other and with known constants, i. e., the equation 
of the locus. If in any particular case the number of equations 
is less than this, one of two things must be true. Either the 
conditions laid down do not force the tracing point to follow a 
definite path, or the student has failed to impose on the co-ordi- 
nates of the point or on the parameters all of the limitations 
imposed by the problem. 

As illustrations of what has been said above consider the solu- 
tions of the following : 

1. A line of fixed length slides along the co-ordinate axes, keep- 
ing one end on each axis. Find the locus of its middle point. 



Analytic Geometry. 



75 



Y 



Fig. 22. 



The point P, {x, y) , has its position 
determined by the two variable quanti- 
ties a and b. If k is the fixed length of 
the line we have as the algebraic trans- 
lation of the restrictions on the move- 
ment of P the following equations : 

a h 

a 2 + b 1 = h\ 
From these three equations we elim- 
inate the two parameters a and b, and 
deduce the single relation 

the equation of the desired locus. 

2. Given a fixed point on a circle and a variable chord through 
that point. Find the locus of the point which divides the chord 

in the ratio — . 

n 

Let be the fixed point, OP any 
position of the 'variable chord, and 
Q the point whose locus we desire 
to find. We are free to locate our 
axes in any position, and in order 
that the work may be as simple as 
possible we select the radius of the 
circle through as the X axis and 
the tangent at as the Y axis. 
The circle then has (r, 0) as its 
center and its equation is 

x 2 — 2rx -\- \f = 0. 
The variation of Q is evidently 

produced by the variation of P along the circle. Let P be (x t , y^) 
and Q (x } i/~) and we have at once as the translation into alge- 
braic language of the limitations on the movement of Q 




Fig. 23. 



///,/•, 



'/ 



m lly 



m + it m -\- it 

two relations embracing x\ \j and the two variable parameters 
x x and y„ one relation less than we need. It is evident that the 



76 Analytic Geometry. 

limitations force Q to trace a definite locus, we must therefore 
have failed to impose one of the limitations of the problem. 
Searching for this omitted limitation we soon see that we have 
not restricted Ca? x , //,) to the circle. Imposing this restriction 
we have our desired third relation 

V— 2rw l + y l * = 0, 

which combined with the two already found enables us to elimi- 
nate m and n and deduce the desired relation between x and y\ 

! — )x —2rx-\r( ! — )y =0 

m / \ m ) 

or, dropping accents, 

in -\- n\ i i ( in + n i 
)x — 2rx + ! — ) // = 0. 

in ) \ hi 

PKOBLEMS. 

1. A, B, C are three fixed points on a" straight line and P a 
variable point subject to the condition 

angle APB = angle BPC. 

Find the locus of P. 
(In this as well as the other problems of this list the student 
will find it well to locate his axes in such a way as to give the 
greatest possible simplicity to the work without destroying the 
generality of the problem.) 

2. Through a fixed point on a circle chords are drawn and 
on each chord, extended, a point P is taken such that OP is twice 
the length of the chord. Find the locus of P. 

3. Subject the general circle 

(x — a)*+(y — 6) 2 = r 2 

to the condition of passing through the two fixed points (x ly y x ) 
and (x , y 2 ) and find the locus of the center. 

4. Find the locus of the point from which a fixed segment AB 
on a ffiven line subtends a right angle. 

5. The two tangents from a variable point to a fixed circle 
make with each other a constant angle. Find the locus of the 
variable point. 

6. The distance of the point P from a fixed point on a circle 
centered at the origin is equal to the slone of the polar of P with 
resoect to the circle. Find the locus of P 

7. A and B are two fixed points. The distance of P from A 
equals the cosine of the angle PAB. Find the locus of P. 



Analytic GEOMETRY. It 

8. The distance of P from its polar witb respect to a given 
circle centered at the origin is equal to the slope of the polar. 
Find the locus of P. 

9. A line moves parallel to its original position. On the line 
a point P is taken so that the distance from P to the Y axis is 
equal to the distance from P to the point where the line meets the 
X axis. Find the locus of P. 

10. A line of constant length slides on the co-ordinate axes, 
keeping one extremity on each axis. Find the locus of its pole 
with respect to a given circle centered at the origin. 

11. The distance of the point P from its polar with respect to a 
fixed circle centered at the origin is equal to the sum of the inter- 
cepts of the polar on the co-ordinate axes. Find the locus of P. 

12. A is a fixed point outside and Q a variable point on the 
circumference of a fixed circle. Find the locus of the point on 
the line AQ whose distance ratio with respect to A and Q is 3. 

13. Find the locus of the poles, with respect to a given circle, 
of a system of parallel straight lines. 

11. A variable line is subject to the condition that it must be 
tangent to 

Find the locus of its pole with respect to 

15. The line joining the point P to a fixed point on the circum- 
ference of a given circle is perpendicular to the polar of P 
with respect to the same circle. Find the locus of P. 

16. Which of the loci deduced above are circles? 



CHAPTER XIII. 
THE GENERAL EQUATION OF THE SECOND DEGREE. 

58. We take up now the study of the genera) 
nature of the conic as represented by the general equation 

PROBLEM AND OF - +1 , , . , * ,. , 

the method second degree, a special case of which 

employed. ^as been studied in the chapter on the circle. 

We shall show that every conic has two axes 
of symmetry and that by making the axes of co-ordinates coinci- 
dent with or parallel to these axes of symmetry all equations of 
the second degree are. reduced to three type forms. These type 
forms will then be investigated in much the same manner as the 
circle was investigated in Chapter XI. 

For the first part of the investigation we shall use the para- 
metric form of the equations of a straight line, developed in para- 
graph 39, which bring into evidence a fixed point (x 19 y x ), the 
direction of a line through the point, and the distance measured 
along the line from the fixed point to a variable point. By put- 
ting the variable point on the conic and rotating the line we 
shall be able to investigate the curve by noting the changing value 
of the distance from the fixed point to the variable point on the 
conic, in much the same manner that the bottom of a lake is investi- 
gated by measurement of its depth at various points. 

59. Consider any point (x t , y x ) and write any 
THE »r» EQUATION. Une through it in the form 

y=y 1 + mr 

where I and m nre the cosines of the angles which the line makes 
with the X and Y axes. To find the distances along the line from 
the point (a? 13 y x ) to the conic we substitute x and y, as given by 
the equations of the line, in the general equation of the second 

degree 

ax" + by 2 + 2hxy + 2gx + 2fy + c=0. 



Analytic Geometry. 79 

This gives us 

[air + tyi 2 + 2hx iyi + 2gx, + 2fy l + c) 

+ 2r[ 2(0*! + %i + #) + w (/i^ + by x + f ) ] 
-fr 2 («Z 2 + 6m 2 + 27Wm)=0 

an equation in r whose roots are the distances from the fixed point 
to the conic. Since the equation is a quadratic there are two such 
distances and we have the theorem : 

A conic is met by any straight line in two points. 

,60. If the point (x x , y x ) is so placed as to be 

ONE chord is midway between the points of intersection 

bisected AT ANY f the line and the conic the two values of / 
point, given by the r equation are equal in value 

and opposite in sign. The necessary and sufficient condition for 
this is the vanishing of the coefficient of the first degree term in r, 
i. e., 

l{ax x + hy 1 + g) + m(hx 1 + ty x + f)= 0. A 

This equation may be satisfied in several ways. First, x x and ^may 
be given any arbitrary values and the equation satisfied by proper 

choice of — , i. e., by giving the proper direction to the line. Note 

that only one value of — will satisfy the equation. From this 

method of satisfying the condition we derive the theorem : 

Through any point in the plane there may be drawn one and in 
general but one chord of a given conic which is bisected at that 
point. 

61. Again, it is always possible to find one and 

CENTER OF A CONIC, only one point which satisfies both of the 
equations 

ax + hy + g = 
hx + ~by-\-f = 

If this point is taken as (x±, y x ) the coefficient of r is zero without 

regard to the value of — ^ i. e., the chord is bisected at (x 19 y ± ) 

L 

no matter what its direction. This gives us the second theorem : 
Every conic has a center of symmetry whose co-ordinates are 



80 Analytic Geometry. 

determined by the equations 

62. Again, let — have a fixed value. It is 

DIAMETERS OF A .,, , ,}. ,, ,.,. . , 

possible to satisfy the condition A by choice 

CONIC. ^ l J 

of (x x , y x ). In fact if we regard — as 

fixed and x ± and y 1 as variable the condition becomes the equa- 
tion of a straight line, which is evidently the locus of the 

middle points of a system of parallel chords with the slope — . 

Such a locus is called a diameter of the conic. It is easy to 
see that all diameters pass through the center of the conic. We 
have now our third theorem : 

The equation of the diameter bisecting the family of chords 

whose slope is — is 

l(ax + hij+g) + m(hx+l)y + f)=0 

63 - An axis of symmetry differs from other 

axes OF symmetry diameters in that it is perpendicular to the 
OF A CONIC. chords it bisects. Given a family of chords 

of slope — the corresponding diameter has the slope 

al -f- hm 



M + bin- 



If this diameter is to be perpendicular to the chords it bisects we 
must have 

W\ / al + 7im \ _ __-, 

l)\ M + hmj 
i. e., aim -\- lim 2 =hl 2 -\- him 

m\ 2 , la — b\m 



ijrvirjT- 1 ^ - 

The roots of this equation are the slopes of systems of chords 
perpendicular to the diameter which bisects them. The equation 
is a quadratic, therefore there are two such systems. The product 
of the two roots is — 1, therefore the two systems are perpen- 
dicular to each other. We have now our fourth theorem : 

Every conic has two axes of symmetry, and these two axes are 
perpendicular to each other. 



Analytic Geometry. 



81 



64. The work of the last article enables us to 

reduction OF the determine the angles which the two axes of 
general equation. symmetry make with the X axis. Let us 
assume that the axes of co-ordinates have 
been made parallel to the axes of symmetry by rotation through 
one of these angles, and let the form of the equation referred to 
these new axes be 

Ax 2 + By 2 + 2Hxy + 2Gx + 2Fy + C = 0. (1) 

Two questions are now before us. This particular choice of axes 
must entail certain values for some of the coefficients of the equa- 
tion or, what amounts to the same thing, special relations between 
them. The determination of these values or relations is our first 
question, and the determination of the distance from the axes of 
co-ordinates to the axes of symmetry is the second. 

Let AB and CD (Fig. 24) be the 
axes of symmetry and let their dis- 
tances from the axes of co-ordinates 
be k and I (equations y — 1 = 0, 
x — k = 6) . The necessary and suffi- 
cient condition that 

x — k = 

may be an axis of symmetry is that 

if any point P 1? co-ordinates (k-\-r, 

y t ), be on the curve, the point Q, co- FlG> 24 ' 

ordinates (k — r, y ± ), shall also be on the curve, i. e., if 

A{k+ry+By*+2H(k+r)y x +2G{k+r)+2Fy x +G=0 
so also A(k— r ) 2 +By 1 2 +2H(k— r)y l ^ r 2G(k— r)+2Fy 1 +C=0 

Subtracting the second from the first and dividing by 4r (Whaf 
right have we to divide by r?) we have 

Ak + G-\-Hy 1 = Q 

an equation of the first degree in y x which is satisfied by the // 
co-ordinate of any point on the conic. This can be true in only 
two ways ; either the general conic must consist of straight lines, 
or the equation last written is not a condition but an identity. 
If the former of these alternatives were true the general equation 
must split up into first degree factors, but it does not; we have 

therefore A k A- G + Hy 1 = 

i.e., Ak + G = and H=0. 




82 Analytic Geometry. 

Similarly treating the other axis of symmetry we have 

Bl + F = and H=0. 

If then the axes of co-ordinates are taken parallel to the axes 
of symmetry of the conic the equation has no term in xy, and the 
coefficients A, B, F, G, are connected with each other and with 
the distances from the axes of co-ordinates to the axes of symmetry 
by the relations 

T A £ 

(The point (fc, I) is evidently the center of symmetry, and its co- 
ordinates might therefore have been found by the method of article 
61.) 

Apply the transformation 

x = x — £ 
A 

F 

and the new axes of co-ordinates coincide with the axes of sym- 
metry of the conic, while the equation reduces to the form 

(1) Ax 2 +By 2 + K = 

in which K denotes the new constant term. If either A or B is 
zero (both cannot be, problem 4, article 33) the above trans- 
formation cannot be made, since in this case the center of sym- 
metry and one of the axes of symmetry are at infinity. The 
simplification of 

Ax 2 + By 2 + 2Gx + 2Fy + O = 

must therefore in this case be accomplished in some other way. 
Let A be the coefficient that vanishes. There is nothing to preven t 
our applying the transformation 

F 

which makes the X axis coincide with that axis of symmetry which 



*i. e., the special choice of axes made at the beginning of this paragraph 
leaves ( 1 ) , not in the general form there written , but in the special form 

Ax* -f- By* — 2kAx — 2lBy + C = 0. 



Analytic Geometry. 83 

is in the finite part of the plane. This transformation reduces the 
equation to the form 

By 2 + 20. jo + L = 
where L is the new constant term. This curve crosses the X axis 

at the point ( — , 0) which is in the finite part of the plane 

1 G 
so long as G does not vanish. We therefore apply the transforma- 
tion 

05 =X — 

2G 
which moves the Y axis to this point of intersection and reduces 
the equation to the form 

(2) B if + 2Gx = 0. 

If G vanishes the equation 

By 2 + 2Gx + L = 
reduces at once without transformation to 

(3) Bf + L = 0. 

We have now succeeded in showing that each and every equation 
of the second degree in two variables may, by a mere transforma- 
tion of co-ordinates, be reduced to one of the types (1), (2), (3) 
above. (3) reduces at once to 



representing two real or imaginary lines parallel to the X axis. 
We may therefore dismiss it from further consideration. (1) and 
(2) remain to be studied. Since all the conies reducible to (1) 
have their centers in the finite part of the plane we shall fre- 
quently refer to them as central conies. 



CHAPTER XIV. 

THE ELLIPSE AND THE HYPERBOLA. 

65 - We lake up now the consideration of type 

determination OF (i) f the preceding chapter, the equation 

FORM. 

Ax 2 +By 2 + K = 0. 
If K vanishes the equation at once reduces to the form 

( yJLx +v =z ^2?) ( V^ — y=^)= o 

and therefore represents two real or imaginary straight lines 
through the origin. If K does not vanish and A, B, and K have ail 
the same sign, the equation cannot be satisfied by any real point 
(sum of three positive or three negative quantities cannot be zero) ; 
and since we are for the present interested only in real loci we 
shall give no further attention to this case. When the signs are 



not all the same divide bv K, put 

1 

equation reduces to the form 

clj/ + Pi/=1 



A 



= a, 



K 



P and the 



This curve meets the X axis in the points ( -^— , 0) and ( — , 0) 

l/a i/a 

and the Y axis in the points (0, -— ) and (0, -), and the lengths 

VP V P 

of the segments determined by the curve on the X and Y axes 

9 9 

(and hence on the axes of symmetry) are — — and —■ — . These 

l/a yp 

values might logically be called the lengths of the axes of the 
curve, but as one or the other of the quantities a, P may be neg- 
ative mathematicians have agreed to define the lengths of the axes 



as the moduli (see appendix E) of these values, i. e., 



y a 



2 
VP 



and in this way avoid the introduction of imaginary lengths. If 
we denote the semi-axes thus defined by a and 1) we have 



2a 



|/ a 



25 i^ 



Analytic Geometry. 



85 



If a and ($ are both positive 
we have 

tt= JL /3 = 1- 

<r If 

and the equation takes the form 



+' 






1 



Solve this equation for y in 
terms of x and the truth of the 
following statements is at once 
evident. As x increases in num- 
erical value, or, to say the same 
thing more technically, as \x\ in- 
creases, \y\ decreases from the 

Y 




Fig. 25. 

value 1) which it has when \x\ 
is zero to the value zero which it 
has when \x\ is a. As \x\ in- 
creases beyond a, \y\ increases 
indefinitely, but since y is imag- 
inary for these values the corre- 
sponding points are not repre- 
sented in the plane. The curve 
therefore lies wholly within the 
rectangle formed by the lines 

x ± a — and y ± 6= 
Careful plotting will show it to 
be of the form here given. It 
is called an ellipse. 



If one, let us say /3 of the 
quantities a, j3 is negative, we 
have 



i- -» 



and the equation takes the form 

jl !L = i 

a b 

Solve this equation for y in 
terms of x and the truth of the 
following statements is at once 
evident. As \x\ increases, be- 
ginning at zero, \y\ decreases 
from the value b, which it has 
when \x\ is zero, to the value 
zero, which it has when \x\ is a. 



Y 





Fig. 26. 

Over this range, however, y is 
imaginary so the corresponding- 
points are not represented in 
the plane. As \x\ increases be- 
yond a, \y\ increases indefin- 
itely. The curve therefore lies 
wholly without the lines 

xzta = 

and extends upward and down- 
ward indefinitely. Careful plot- 
ting will show it to be of the 
form here given. It is called an 
hyperbola. 



86 



Analytic Geometry. 



In plotting this diagram a 
was assumed greater than b. 
If a is less than b> the conic is 
turned along the other axis. 
The question of size is evidently 
the important one in distin- 
guishing between the axes and 
they are therefore spoken of as 
major and minor. In the devel- 
opment of the theory of the 
ellipse we shall assume that a 
denotes the length of the semi- 
major axis. 



In plotting this diagram f3 
was assumed negative. If a 
is negative the conic is turned 
along the other axis. The im- 
portant question here is not one 
of size but of the character (real 
or imaginary) of the points in 
which the conic meets the axes. 
The axis met by the conic in 
real points is called the trans- 
verse axis, the one met in im- 
aginary points is called the 
conjugate axis. In the devel- 
opment of the theory of the 
hyperbola we shall assume that 
a denotes the length of the 
semi-transverse axis. 



66. 



EARLY GEOMETRIC 
DEFINITIONS. 



These curves were well known to geometri- 
cians before the invention of analytic geom- 
etry, and each of them had its geometric defi- 
nitions. Two of these are as follows : 
(A) An ellipse An hyperbola 

is the locus of a point the 



sum 



difference 



is 



of whose distances from two fixed points, called the foci 

constant. 

(B) An ellipse An hyperbola 

is the locus of a point whose distance from a fixed point divided by 

its distance from a fixed line is a constant 

less greater 

than unity. 

The fixed point is called the focus, the fixed line the directrix, and 

the constant ratio the eccentricity. 

If we attempt to deduce the equations of these curves directly 
from the definitions just given the resulting forms will depend 
upon the choice of co-ordinate axes. In deducing the equations 
from definition (A) we choose the line joining the two foci as the 
X axis and the perpendicular bisector of the segment between the 



Analytic Geometry. 



87 



foci as the Y axis. Then if the distance between the foci be taken 
as 2c the foci are ( — c, 0) and (c, 0) and our definitions lead at 
once to the equations 



V (5 + cT + if 



+ V {x — cY 



a' 



2k 



l/ (a + cY + 1/ 



V(x — e) 2 + y=2k 



where 2k is the constant 
sum | difference 

mentioned in the definition. Eationalize and reduce and each of 
the forms leads to the single equation 

2 



,r 



+ 



J/_ — 



= 1 



¥ k 2 — c 2 
If we compare this with our standard forms Ave have 



a- 
b 2 



k 2 
-k 2 — & 
= a 2 — & 



— l 2 = 

= a* — c 

c 2 = a 2 +6 2 



k 2 

k 2 — c 2 

2 n2 



and we have as foci the two real points 



(Va 2 -l 2 ,0) 

(—ya 2 — V-, 0) 



(yq » + B», ) 

(— ya 2 -\-~b 2 , 0) 



Note that the foci w r ere assumed on the X axis and the resulting 
equation identified with one for which the X axis is the 

major transverse 

axis of symmetry. Hence we may say that the foci are two real 
points on the 

major transverse 

axis of the conic and symmetrically situated with respect to the 
other axis. 

If on the other hand, the foci are assumed on the Y axis the 
resulting equation is 



2 2 

x , y 



G tC 



Identifying this with the same equations as before, we have 



l 2 = k 2 

= h 2 — c 2 
c 2 =h 2 — a" 



V- = k 2 

a 2 = k 2 — c 2 

= — l) 2 — & 
c 2 = — a 2 — b' 



88 Analytic Geometry. 

and we have as foci the two imaginary points 



(0, yv — a 2 ) 



(0,— V& 2 — a 2 ) 



(0, V— a 2 — h 2 ) 



(0, — V-a 2 — & 2 ) 



Note that in this case the foci are assumed on the Y axis and the 
resulting equation identified with one for which the Y axis is the 

minor conjugate 

axis of symmetry. Hence we may say that the foci are two imagin 
ary points on the 

minor conjugate 

axis of the conic and symmetrically situated with respect to the 
other axis. 

But since either assumption leads, when c and k are properly 
determined, to the same equation 

a 2 ~ b 2 

it follows that the conic represented by this equation (i. e., any 
central conic) has four foci, two real on the 

major transverse 

axis, and two imaginary on the 

minor conjugate 

axis. Hereafter when the foci are referred to it is understood that 
the reference is to the real foci unless both are mentioned. 

In deducing the equations from definition (B) we choose the 
directrix as the Y axis and the perpendicular let fall upon it 
from the focus as the X axis. We denote the distance from the 
focus to the directrix by cl and the eccentricity by e. The definition 
then leads at once to the equation 



V(g — d) 2 + y 2 =p 
x 
or on reduction 

x 2 ( 1 — e 2 ) + y 2 — 2dx + d 2 = 0. 

The absence of the xy term shows that the axes of co-ordinates are 
parallel to the axes of symmetry, and the absence of the y term 
shows that the X axis coincides with an axis of symmetry. But 
the presence of the x term shows that the Y axis does not coincide 
with an axis of symmetry and a transformation of co-ordinates 



Analytic Geometry. 



8:) 



must be made before we can compare our equation with 1 lie stand- 
ard forms. We therefore bring the Y axis into coincidence with 
the other axis of symmetry by the substitution 



x — x 



d 



where 



is the distance from the directrix to the axis of 



symmetry to which it is parallel. The transformation gives an 
equation which reduces finally to 

y 



<r 



+ 



dV 



1, 



(1— e 1 ? 1 — e 2 

an equation of the desired standard form. 

For the ellipse e is less than 
unity and the coefficients of 
both x 2 and y 2 are positive. 
Comparing our present equa- 
tion with the standard form we 
have 

(1-V) 2 

dV 



whence 



b 2 = 

Jl = 

a 2 



= 1 — e 2 
~b 2 



Again, replacing e by 



and 



e 2 by — we have, on solv 



ing for d 



d 



For the hyperbola e is greater 
than unity and the coefficient 
of x 2 is positive while that of 
y 2 is negative. Comparing our 
present equation with the 
standard form we have 

rTe 1 





a — 


a-e 2 r 






-b 2 - 


_ dV 






CI - e) 




w iience 


_£ 


= l-e 2 






a 

V 








a 2 + b 2 c 




£ 




a a 




Again, 


replacing e by — 


and 






a 




1 — e 2 


by — 


72 

— we have 

2 

a 


on 


solving 


for d 








d- 


c 





The quantity c will hereafter be called the linear eccentricity. 



90 Analytic Geometry. 

An investigation of the significance of the double sign of d 
leads to interesting results. 

To take the positive sign is to assume that the focus is on the 
right of the directrix, while the distance which the Y axis must be 
moved to pass from coincidence with the directrix to coincidence 
with the axis of symmetry is 

positive (e less than unity) and negative (e greater than unity), 
greater than d. 

To take the negative sign is to assume that the focus is on the 
left of the directrix, while the distance through which the Y axis 
is moved is 

negative positive 

with the same numerical value as before. 

In other words, if we start with a focus on the right of a 
directrix and move the Y axis a certain distance to the 

right left 

we reduce the equation to a certain form. If we start with a 
focus on the left of a directrix, we have a different set of axes 
and a different equation. But when we move the Y axis the 
same distance as before to the 

left right 

we reduce the equation to the same form as before, i. e., the 
two equations given by the two signs of d represent the same 
curve, but referred to different systems of co-ordinates. ' 

Evidently therefore the two signs of d correspond to two foci 
and two directrices. Evidently also the directrices are sym- 
metrically situated with respect to one axis of symmetry of the 
conic and cross the other axis at points 

without within 

the segment determined on that axis by the foci. It is not difficult 
to show that these points are also 

without within 

the segment determined on the axis by the intersection of the 
axis and the conic. 

There are of course a pair of directrices corresponding to the 
pair of imaginary foci. These are however imaginary, a state- 
ment whose proof will follow at once from the solution of problem 
8, article 71. 



Analytic Geometry. 



91 



67. Defini- 

MECHANICAL tion (A) 

constructions. leads to a 
simple me- 
chanical method of construct- 
ing an ellipse. Fasten at the 
two foci the two ends of a cord 
whose length is the constant 
sum of the focal distances of 
the tracing point. Draw the 
cord to one side with a pencil 
and draw the pencil along keep- 



Definition (A) leads to a 
simple mechanical method of 
constructing an hyperbola. 
Make a ruler of the form shown 
in Fig. 28 with the center of 
the opening P on the straight 
edge AB extended. By means 
of this opening P pivot the 
ruler at one focus F x . To the 
other end, B, of the ruler fasten 
one end of a cord shorter than 
the ruler by an amount equal 




(% 



>3 



Fig. 2' 




mg the cord tightly drawn. 
The resulting curve evidently 
satisfies the definition of an 
ellipse. 



Fig. 28. 

to the constant difference of 
the focal distances of the trac- 
ing point and fasten the other 
end of the cord at the other 
focus F 2 . Rotate the ruler 
about F x , keeping the cord 
pressed tightly against the 
ruler by a pencil. The result- 
ing curve evidently satisfies 
the definition of an hyperbola. 
The other branch is drawn by 
reversing the apparatus. 



PROBLEMS. 

1. Let the lengths and positions of the axes of an ellipse and 
an hyperbola be given. Deduce geometric constructions for the 
foci and the directrices. 



92 Analytic Geometry. 

2. Find the eccentricity, lengths of axes, location of foci and 
location of directrices of the following central conies: 

Zx 2 — 2\f +1 = ±x- + 2y 2 = 12 

— 9x 2 + 10y 2 = 1 11 ±x- + y* = 1 

3. The following conies are assumed to have their centers at the 
origin and their axes of symmetry as axes of co-ordinates Find 
their equations. 



w H 


a = £. 


(2) a=5, 


c = 2. 


(3) cl = 2, 


e = 4. 


(4) a = 3, 


b = 2. 


(5) d = 2, 


6 = 1 


(6) c = 4, 


a = S. 



4. Express the distance between the two directrices in terms of 
a and e and show that for the ellipse it is greater than 2a, and for 
the hyperbola less than 2a, 

5. Let the length of the transverse axis of an hyperbola be con- 
stant and let the eccentricity increase indefinitely. Show that 
under these conditions the directrix tends to coincide with the 
conjugate axis. 

6. To what limiting form does the ellipse tend as a tends to h ? 
What is the limit of the eccentricity? 

7. When a tends to h the hyperbola tends to the limiting form 
represented by 

x 2 — tf = ~b 2 

which is called an equilateral hyperbola. What is its eccentricity ? 

8. Show that as an ellipse tends to a circle the distance of the 
directrix from the center tends to infinity. 

9. If in any conic there be erected at the focus a perpendicular 
to that axis of symmetry which passes through the focus, the 
distance between the two points in which this perpendicular meets 
the curve is called the length of the parameter of the conic. This 
length is usually denoted by 2p. If the conic is a central conic 
referred to its axes of symmetry as co-ordinate axes, the parameter 
might be defined as the double ordinate through the focus, or as 
that portion of the line x -\- c=.0 or x — c = included between 
its intersections with the conic. Show that for the ellipse the 
semi-parameter p is a third proportional to the semi-axes. Show 
also that p = a (1 — e 2 ) . 

10. Determine the corresponding values of p for the hyperbola. 

11. Move the axis of Y so that it shall become the tangent ar 
the left hand vertex of the conic. (The vertices of a conic are its 



Analytic Geometry. 93 

intersections with its axes of symmetry.) The equation of the 
ellipse then reduces to the form 

if = ., (2dX — ./' ) 

a 

or if = 2 px (1 — — ) . 

2 a 

Let a and b increase indefinitely, but in such a way that the ratio 

_(=p) remains constant. What is the limiting value of s? 
a 

What is the limiting form of the second of the two equations just 
given ? 

12. Make a similar investigation for the hyperbola. 

13. Compare the limiting forms developed in the last two prob- 
lems with equation (2), article 64, and note that the assumptions 
just made concerning a and & have the effect of moving the center 
of the conic to infinity, and therefore correspond to the assump- 
tions which led to equation (2) . Hence we may state the theorem : 
The conic represented by form (2) is the limiting form of both 
the ellipse and the hyperbola as the eccentricity tends to unity. 

14. The lines joining any point on a conic to the foci are called 
focal radii. Denote their lengths by r and r, let the axes of co- 
ordinates be the axes of symmetry, and show that for the hyperbola 

r = ex — a, , r = ex + a. 

15. What are the corresponding values for the ellipse? 

68 - Applying to the special equation of the 

diameters. second degree 

ax 2 + Pif = 1 
the general formulae for the equation of a diameter of a conic 
developed in article 62, we find that the equation of the diameter 
bisecting chords of slope m is 

ax + fimy = i.e. 

b 2 x-\-a 2 my = b 2 x — a 2 my = 

Given two diameters 

y = m x x and y = m 2 x 
the work just done shows that the necessary and sufficient condi- 
tion that the second diameter shall bisect all chords parallel to the 
first is 



m x m 2 = — ~ i.e., 



72 

m 1 m 2 = — \ 
a 



a 



v- 



94 Analytic Geometry. 

From the form of this condition it is evident that if the second 
diameter bisects all chords parallel to the first, the first also 
bisects all chords parallel to the second. Such a pair of 
diameters are said to be conjugate diameters. 
Let 

y = m x x and y = m 2 x 

be a pair of conjugate diameters of the conic 

ax 2 + Pif = 1 

and let the points in which they meet the conic be (x 19 y x ), {x 2 , y 2 ) , 
(— #!, — y t ), {—x 2 , — y 2 ) then 

m 1 = — and m 2 = — 
and therefore since the diameters are conjugate 



a 

x,x 9 ft 



ViVz _ (i) 



1" 2 



Also since the points are on the conic 

ax 2 + P Vl 2 = 1 (2) 

and ax. 2 + Py 2 = 1 (3) 

Solve (2) and (3) for x x and y 2 and substitute in (1) whence we 
have 

Substitute this value of x 2 in (1) and we have 

If expressed in terms of the lengths of the axes this becomes 







y 2 — _)_ x-i 
a 



o 
a 



where the upper sign of x is paired with the upper sign of y. In 
deciding upon the arrangement of signs for the hyperbola the 
student must remember that the negative sign is associated with & 

and that — is equal to — i. 



Analytic Geometry. 05 

PROBLEMS. 

1. Show that as one of a pair of conjugate diameters of a 
conic tends to coincide with one axis of symmetry the other tends 
to coincide with the other axis of symmetry. 

2. Show that if one of two conjugate diameters of an hyperbola 
meets the curve in real points the other meets it in imaginary 
points, and conversely. 

3. Show that two conjugate diameters of an hyperbola are in 
the same quadrant, and find the angle between them in terms of 
the semi-axes and the slope of one of the diameters. 

4. Make a similar investigation for the ellipse. 

5. Let two conjugate diameters of a conic meet it in_the 

points (x lf ?a) and (a? 2 , y 2 \ i.e. in the points (x 1} y x ) and (<\/— Vi> 

— \ H Xl )> an( ^ ^ ^ ie distances of these points from the center 
be denoted by d 1 and d 2 . Then 

7 2,72 2 , 2 , ft 2 , Ct, 2 

d l + d 2 — x x + y x ^ y t + — x x . 

a p 

— ax i + Plh 4. ax ? + Pi*? 

a (3 

— 1.1 

In other words the sum of the squares of these distances is con- 
stant, i. e., remains unchanged for all positions of the conjugate 
diameters. In the ellipse both these distances are real, but in the 
hyperbola one or the other is imaginary. In consequence of this 
mathematicians agree, as in the case of the axes, to define the 
length of a semi-diameter of a conic as the modulus of the distance 
from the center of the conic to the point of intersection of the 
diameter and the conic. It is usual to denote the lengths of a pair 
of conjugate semi-diameters by a and Z>'. With the above defini- 
tion in mind, show that the work just done is equivalent to the 
proof of the theorem : 

The sum The difference 

of the squares of the lengths of a pair of conjugate semi-diameters 

of an 
ellipse hyperbola 

is constant and equal to the 
sum difference 

of the squares of the lengths of the semi-axes. 

6. Find the sines and cosines of the angles made by a pair of 
conjugate diameters with the axes of an hyperbola, expressing 
them in terms of a, ~b, a, o', x l9 y ± . 



96 Analytic Geometry. 

7. Make a similar investigation for the ellipse. 

8. Snow that the sine of the angle between any pair of conjugate 

diameters of an ellipse is ~ t . 

ab 

9. Make a similar investigation for the hyperbola. 

10. By aid of the values deduced in problems 6 and 7 find the 
equations of the ellipse and hyperbola referred to a pair of con- 
jugate diameters as oblique axes of co-ordinates, and show that the 
equations reduce to the forms 



a ' o" 



2 2 

* J/_= I 

'2 7/2 - 1 - 

a o 



69 - If any point on a central conic be joined to 

supplemental the extremities of any diameter two chords 
chords. are formed which are called supplemental 

chords. If the diameter is the 
major transverse 

axis they are called principal supplemental chords. Let (x x , y x ) 
be any point on a central conic and (a? 2 , y 2 ) one of the extremities 
of any diameter. Then if the slopes of the two supplemental 
chords determined at (a? 1? y x ) by this diameter are m x and m 2 

Vx — Vi ?/i + V* 

m 1 =r m, — — ; 

and. m x m % = — 2 r 2 • 

tC'j tA/2 

But the conditions which must be satisfied in order that (x ly y x ) 
and (a? 2 , i/ 2 ) may be on the curve give on subtraction 

Vi—V* a 

«*■! "^2 A^ 

hence m x m 2 = — . 

1 2 )8 

i. e., the condition which must be satisfied by the slopes of two 
supplemental chords is the same which must be satisfied by the 
slopes of two conjugate diameters. Therefore if one of two sup- 
plemental chords is parallel to a diameter, the other is parallel 
to the conjugate diameter. 

PROBLEMS. 

1. Given a central conic and a diameter, construct with ruler 
and compass the conjugate diameter, first from the definition of 
conjugate diameters and second by aid of a pair of supplemental 
chords. 



Analytic Geometry. 97 

2. Given a central conic, find its center with ruler and compas3. 

3. Given a central conic and a diameter, construct by aid of 
this diameter a pair of supplemental chords perpendicular to each 
other. 

4. Given a central conic construct its axes with ruler and 
compass. 

PROBLEMS. 

70. l. Following the analogy of the work done 

tangents and in article 49, show that the condition that 

normals. the line 

y = mx + h 
shall be tangent to the conic 

ax 2 + p y a =i 



is h=±\/a 2 m 2 + l) 2 h = ± \/a 2 m 2 — l) 2 

Note that after the condition of tangency has been imposed on 
the equation 

y = mx + h 

it contains only one arbitrary parameter. It follows therefore 
that the tangents which can be drawn to any central conic form 
a single infinity of lines. 

2. Following the method employed in article 50, develop the 
equation whose roots are the slopes of the tangents from (x ± , y x ) 
to a central conic. Hence show that if the point is not on the 
conic, two tangents to the conic can be drawn through it. 

3. Following the method of the latter part of article 50, show 
that the equation of the tangent to the conic 

at the point (x x , y x ) on the conic is 

ax x x + Py x y = 1 

State the equations also in terms of a and J> for the ellipse and 
hyperbola. 

4. Find the equations of the normals at the same point. 

5. Show that the lengths of the sub-tangent and sub-normal for 

the point (x x , y x ) are 

1 — a.7-, 2 -, a 

L and —a 1 !. 

ax x p 

6. Examine the following lines for cases of tangency to 

<£ + £ = ! or «L-1L =1 

9 4 9 4 

2/ + 3 j* — 4 = y = 2 

x — 4ty + 3 = 5x — 3y + 3 = 

3?/ — 4a?— 12 = # — 3i/ + 5 = 



98 Analytic Geometry. 

7. Show that the two tangents which may be drawn to any 
central conic from a given point are real or imaginary according 
as the point is outside or inside the conic. 

8. Find the equations of the tangents from the point (3, 2) 
to the conies of problem 6. 

9. Show that the tangent and normal at any point bisect the 
angles formed by the focal radii at that point. 

10. From the results of the last problem derive a geometric 
construction for the tangent and normal at any point on a central 
conic. 

11. Find the locus of all points from which the two tangents 
to a central conic are perpendicular, and construct the locus with 
ruler and compass. Is the construction always possible in the case 
of the hyperbola? 

12. Find the locus of the feet of the perpendiculars let fall from 
either focus on the tangents to an ellipse. 

( Take y = mx + h equation of straight line. 

h= ±ya 2 m 2 -f- b 2 condition of tangency. 

y = — ( — ) (x — c) perpendicular to first line 

m 

through focus. From these equations eliminate h and m, and 
derive the desired locus in the form 

y 2 = x(c — x)± V & 2 (c — x) z + b' A y' 2 . 

Rationalize, replace b by its value in terms of a and c, arrange 
the equation according to powers of c, and the equation reduces 
to 

(a) 2 + y*^a 2 y(w 2 + y 2 —2cx + <?)=0 

The locus is evidently degenerate, representing a circle concentric 
with the conic and a pair of imaginary lines intersecting at the 
focus. The question whether the real and imaginary parts of this 
locus correspond to the perpendiculars let fall upon real and im- 
aginary tangents is left for the investigation of the student.) 

13. What changes must be made in the above investigation in 
order to make it hold for the hyperbola ? 

14. Show that the product of the two perpendicular distances 
from the two foci of a central conic to the tangent at (x ly y ± ) is 

— b*a 2 e 2 x 1 2 + a*b± 
b 4 x ± 2 + tfy 2 

Eliminate y x by virtue of the fact that (x ± , y x ) is on the conic, 
replace e by its value in terms of a and ~b and thus reduce the value 
of the product to b 2 . 

15. Show that the tangents at the extremities of any diameter 
are parallel to the conjugate diameter. 



Analytic Geometry. 9£ 

1G. By aid of the last problem and problem 8, article 68, show 
that the area of the parallelogram formed by the tangents at the 
extremities of any pair of conjugate diameters of a central conic 
is independent of the position of the diameters and equal to the 
area of the rectangle formed by the tangents at the extremities of 
the axes. 

71. Follow the methods used in the investiga- 

poles and polars. tion of poles and polars of the circle, and 
solve the following : 

PROBLEMS. 

1. Find the equation of the polar of (x 1} y x ) with respect to the 
conic 

ax 2 + Pif = 1. 

2. Find the co-ordinates of the pole of 

ax -\-~by + c = 
with respect to the conic 

ax + /V = 1. 

3. Show that if the pole of a given line with respect to a given 
central conic is on the conic the polar is the tangent at the pole. 

4. Show that if a point lies on its own polar with respect to a 
central conic, it lies also on the conic. 

5. Show that if the polar of a point A passes through B the 
polar of B passes through A. 

6. Given any pair of conjugate diameters of a central conic show 
that the polars of all points on the one are parallel to the other. 

7. Give geometric constructions for poles and polars in the case 
of both ellipse and hyperbola. 

8. Show that the directrices are the polars of the foci. 

9. If it can be shown that any investigation does not depend 
on the rectangularity of the axes, the results of the investigation 
hold good for oblique axes. Show in this way that the form of the 
equation of the polar developed above holds good so long as the 
central conic is referred to a pair of conjugate diameters as co- 
ordinate axes. 

72. If in the equation of the tangent at any 

asymptotes. point ((D u y ± ) on a central conic we substitute 

the value of y ly deduced from the fact that 

( x i> Vi) i s a point on the conic, we have a form which reduces 

at once to 



ax 






100 Analytic Geometry. 

The double sign before the radical arises from the fact that there 
are two values of y 19 and hence two tangents, corresponding to a 
single value of x x . If now the value of x x is allowed to increase 
indefinitely, i. e., if the point of tangency is allowed to recede 
indefinitely from the origin, this equation of the tangent tends to 
the limiting form 

7/ cue ± %V$V = 
or lx ± lay = l)x ± ay = 

In other words the tangents at infinity to the ellipse are a pair 
of imaginary lines intersecting in the center of the ellipse, and for 
the hyperbola a pair of real lines intersecting in the center of the 
hyperbola and forming the diagonals of the rectangle on the axes. 
Tangents at infinity to any central conic are called its asymptotes.* 

PROBLEMS. 

1. Find the angle between the two asymptotes of a central conic. 

2. Show that the asymptotes of an equilateral hyperbola are per- 
pendicular to each other. From this fact the equilateral hyper- 
bola is sometimes called a rectangular hyperbola. 

3. Show that any asymptote regarded as a diameter is its own 
conjugate. 

4. Show that any two conjugate diameters of an hyperbola are 
separated by an asymptote. 

5. Show that if the asymptotes of an hyperbola be taken as a 
pair of oblique axes the equation of the hyperbola reduces to the 
form 

_ a 2 + V- _ cr + b 2 

xy — — - — or xy — — 

4 4 

according to the choice of positive directions of the new axes. 

6. Show that the equation of the tangent at any point (a? 1? y x ) 
on the hyperbola has the form 



*This definition might be made a general definition of an asymptote to 
any curve were it not for the fact that in special cases the line at infinity 
is itself a tangent, and some mathematicians prefer to exclude it from the 
list of possible asymptotes. 



Analytic Geometry. 101 

JL + J/_ = 2 
«i .'A 

if the asymptotes are the axes of co-ordinates.* 

7. Show that the segment of the tangent included between the 
asymptotes is bisected at the point of tangency.t 

8. Show that the product of the intercepts of the tangent on the 
asymptotes is independent of the position of the tangent and equal 
to the sum of the squares of the lengths of the semi-axes. 

9. Show that the area of the triangle formed by the asymptotes 
and any tangent is independent of the position of the tangent and 
equal to the product of the lengths of the semi-axes. 

73. Closely associated with the hyperbola 

THE CONJUGATE 

HYPERBOLA. #! \L~-\ 

2 7.2 ■ L 

a b 
there is a second hyperbola 

2 2 

x , y_ 



_ <i_ + 4L. = 1 

a 



.2 J2 



which is turned along the Y axis in place of the X axis and has the 
transverse and conjugate axes of the original hyperbola as con- 
jugate and transverse axes. It is called the conjugate of the 
original hyperbola and plays an interesting part in the theory of 
conjugate diameters. 

PROBLEMS. 

1. Show that if a diameter meets an hyperbola in real points 
its conjugate meets the conjugate hyperbola in real points and 
conversely. 

2. Show that an hyperbola and its conjugate have the same 
asymptotes. 



*The student will find it simpler to deduce the equation of the tangent 
directly from the equation 

a- 4-5 2 

x v-—i — 

than to deduce it by transformation of 

resi __ VVx = i 

a 2 b* 

fRemember that while the formula for distance presupposes rectangularity 
the formulae for division of a segment in a given ratio do not. 



102 



Analytic Geometry. 



3. Show that if a diameter meets an hyperbola in imaginary 
points it meets the conjugate hyperbola in real points whose co- 
ordinates are the moduli of the co-ordinates of the imaginary 
points in which it meets the original hyperbola. 

4. Show that the asymptotes separate the diameters of an 
hyperbola into two groups (whose slopes are respectively less and 

greater than--) one meeting the original hyperbola in real points 

a 
and the other meeting the conjugate hyperbola in real points. 

74. The two circles having the center of the 

the auxiliary ellipse as centers and the major and minor 
circles. axes of the ellipse as diameters are called the 

major and minor auxiliary circles. The 
practical importance of these circles in the theory of the 
ellipse is largely due to the fact that they serve to connect 
the ellipse with the circle, a curve for which we have a full 
and for the most part simple geometric treatment. The corre- 
sponding curves for the hyperbola are a pair of equilateral hyper- 
bolas, and the lack of a corresponding geometric treatment for 
equilateral hyperbolas renders the analogues of the following prob- 
lems of small practical importance : 



PROBLEMS. 

1. Show that if a point on the ellipse and one on the major 
auxiliary circle have equal abscissas their ordinates are in the 

ratio — . 
a 

2. Given an ellipse and its major 
auxiliary circle, divide the major 
axis into a number of equal parts 
and on these parts construct pairs 
of rectangles of which AD and AF 
are a sample pair. From the last 
problem the areas of any such pair 

are evidently in the ratio — . But 

a 
the areas of the ellipse and the 
major auxiliary circle are evidently 
twice the limits of the sums of these 
rectangles as the number of parts 
into which the major axis is divided 
tends to infinity. Show from this FlG . 29. 

that the area of the ellipse is irab. 




Analytic Geometry. 10:3 

3. Find the areas of the ellipses of problems 2 and 3, article 
G7. 

4. If through any point D on an ellipse the ordinate be drawn 
and extended till it meets the major auxiliary circle in the point 
/•'. then the angle FOX is called the eccentric angle of the point D. 
Show that if (x x , y x ) is any point on an ellipse and ^the eccentric 
angle of that point 

x 1 = a cos $ l Vi = ^ sin l 

5. In onr investigation of the sub-tangent it was found that 
the length of the sub-tangent corresponding to the point (x 19 y t ) 
did not depend on either b or y t . It follows therefore that if a 
family of ellipses is constructed with the same major axis but 
with different minor axes, an ordinate erected at any point in 
the major axis will cut this family of ellipses in a series of points 
having the same sub-tangents, i. e.. the tangents at all these points 
meet at the same point on the major axis. This family of ellipses 
includes the major auxiliary circle. Hence derive a method of 
drawing a tangent at any point on an ellipse with ruler and 
compass. 



CHAPTER XV. 



75. 
DETERMINATION 
OF FORM. 



THE PARABOLA. 

We take up now the consideration of type 
(2) of Chapter XIII, the equation 

By 2 +2Gco = 0, 

in which neither B nor G is zero. Solve fovy, put ~=c, and 

we have 

y 2 — 2C00. 

If c is positive the values of y are imaginary when % is negative, , 

zero when x is zero, and real when % is positive. As x increases 

from zero to plus infinity y also 

increases to plus infinity for one 

set of values and decreases tc minus 

infinity for the other set. The curve 

therefore lies wholly to the right of 

the Y axis, passes through the origin 

and extends upward and downward 

indefinitely. Careful plotting will 

show it to be of the form here given. 

It is called a parabola. If c is 

negative, all that has been said 

holds except that the curve lies to the fig. 30. 

left of the Y axis. 

76. The parabola, like the ellipse and the hy- 

early geometric perbola, was known to geometricians before 
definition. the discovery of analytic geometry. One of 

its geometric definitions is as follows: The 
parabola is the locus of a point moving in such a way that 
its distance from a fixed point, divided by its distance from a 
fixed line, is equal to unity. As in the case of the ellipse and 
hyperbola the fixed point is called the focus, the fixed line the 




Analytic Geometry. 



105 



directrix and the constant ratio the eccentricity. Selecting for 
axes of co-ordinates the directrix and the perpendicular upon it 
from the focus, and denoting the distance from the focus to the 
directrix by d, we deduce at once from the definition just given 
the equation 

The form of this equation shows at once that the X axis is an axis 
of symmetry, that the Y axis is parallel to the other axis of sym- 
metry, and that the vertex of the curve is at the point ( — , 0) . 

Moving the Y axis parallel to itself till it passes through this 
vertex we reduce the equation to the form 

y 2 = 2dco 
which corresponds to the form of the previous article. The defini- 
tion of the curve given above shows also that the distance d equals 
the semi-parameter p, as defined in article 67, problem 9. We 
have therefore the following relations between the constants we 
have employed : c = d=p. 

By means either of the geometric definition of the parabola or 
by consideration of the work done in problems 11, 12, 13, article 
67, we see at once that the parabola is the limiting form of either 
the ellipse or hyperbola as the center tends to infinity under the 
restriction p = constant, or as the eccentricity tends to unity. 
The student will find it both interesting and profitable to derive 
theorems for the parabola by an examination of the limiting form 
of the corresponding theorems for the ellipse or hyperbola. 



77. 

MECHANICAL 
CONSTRUCTION 



To trace the 
parabola mechan- 
ically we place 
one edge of a rect- 
angular board GE 
against the directrix AB, fasten one 
end of a cord equal in length to GD at 
D and the other end at the focus F, and 
slide the board along the directrix, keep- 
ing the cord pressed against the edge 
GD by a pencil point P. The point P 
will trace the parabola, for in every 
position we have PF = PG. 



B 




K 


. C 


\^^ 


j> 


'// 




A 


\f 



Fig. 31. 



106 Analytic Geometry. 

PROBLEMS. 

1. Show that the vertex of a parabola is midway between the 
focus and the directrix. 

2. Find the focal radius of any point (x ly y x ) on the parabola 

y 2 = 2px 

3. Find the foci, parameters, and directrices of the following 
parabolas. 

y 2 —10x y 2 = — 4# 

y 2 = aw + Z) y 2 = — 4# -f- 5 

% 2 — 4:y % 2 =3y -L. i 

if — x + 1 = x 2 + 2y — 1 = 

2/2 = 3^ _|_ 4 3^2 _}_ 4^ — 2 = 

4. Write the equations of the following parabolas : 

x — 2 = is the directrix, and (3, 4) the focus. 

(3, 4) is the focus, axis of symmetry is parallel to the Y 

axis, curve opens upward, parameter is 6. 
(2, 3) is the vertex, (1, 3) the focus. 

78 - The equation of the diameter of the para- 

diameters. bola may b e regarded as a special case of the 

general equation developed in article 62, or 
it may be found directly by determining the locus of the middle 
points of a system of parallel chords. If the latter method is 
adopted, the student should remember that in any locus problem 
the thing sought is the equation which determines the movement 
of the tracing point. If then, as in the present instance, there is 
developed in the course of the discussion an equation of the type 

y=k 

(where Jc is a constant and y is a co-ordinate of the tracing point) 
this equation in itself determines the path of the tracing point, 
and is therefore the equation desired. 

PROBLEMS. 

1. Write the equation of the diameter of the parabola y 2 = 2px 
bisecting the chords of slope m. 

2. Given a parabola y 2 = 2px, and a diameter i/ = 4a, what is 
the slope of the chords which it bisects? 

3. Show that all diameters of a parabola are parallel to the axis 
of symmetry ; first from the form of the equation of the diameter, 
and then from the location of the center. 

4. Why do we not develop for the parabola the theory of con- 
jugate diameters and supplemental chords? 



Analytic Geometry. 107 

79. PROBLEMS. 

TANGENTS AND 

normals. !• What condition must be satisfied by the 

coefficients of the line 

y = mx + h 

in order that it may be tangent to the parabola 

xf = 2px1 

2. Develop the equation whose roots are the slopes of the tan- 
gents from the point (x lf y x ) to the parabola 

y 2 = 2px 

Hence show that if the point is not on the conic two tangents to 
the conic can be drawn through it. 

3. Show that the equation of the tangent to the parabola 

y 2 — 2pX 

at the point (x ± , y x ) is 

yy 1 =p(x + x 1 ) 

4. Find the equation of the normal at the same point. 

5. Find the length of the sub-tangent and show that it is 
bisected at the vertex. 

6. Show that the subnormal is constant and equal to the semi- 
parameter. 

7. Hence deduce geometric constructions for the tangent and 
normal at any point of a parabola whose focus and axis are given. 

8. Show that the two tangents which may be drawn from any 
point to a parabola are real or imaginary according as the point 
is without or within the parabola. 

9. Show that the tangent and normal at any point bisect the 
angles formed by the focal radius of the point and the diameter 
through the point. Show also that this is a special case of problem 
9, article 70. 

10. Hence deduce geometric constructions for the tangent and 
normal at any point on a parabola whose focus and axis are given. 

11. Note that problem 9 is equivalent to the statement that the 
tangent makes equal angles with the axis of symmetry and the 
focal radius through the point of tangency, and hence deduce a 
geometric construction for the tangent and normal. 

12. Find the locus of the points from which two perpendicular 
tangents may be drawn to a parabola. 

13. Find the locus of the foot of the perpendicular let fall from 
the focus upon a variable tangent to the parabola, and show that it 
is degenerate and consists of the tangent at the vertex and a pair 
of imaginary lines. 



108 Analytic Geometry. 

14. Hence deduce a geometric construction for the tangent at 
any point of a parabola whose focus and axis are given. 

15. Show that problems 12 and 13 are special cases of the results 
obtained in problems 11 and 12 of paragraph 70. 

16. Show that the perpendicular distance from the focus to any 

tangent is equal to~\/2pr, where r is the focal radius of the point 

of tangency. 

17.. Show that the tangent at the extremity of any diameter is 
parallel to the chords bisected by that diameter. 

18. Deduce the equation of the parabola referred to any diam- 
eter and the tangent at the extremity of that, diameter as 
oblique axes, and show that it reduces to the form 

y 2 = 2p'x, 

where p is a new constant. What is the value of p in terms of p 
and the angle between the new axes ? 

19. The slope of the tangent to the parabola y 2 = 2px at the 

point (#!, f/i) is — . Since (x 17 y ± ) is on the conic this may be re- 
_ V\ 

duced to JP_ , Show that when the point of tangency removes 

\2i\ 

toward infinity the tangent tends to parallelism with the axis of 
symmetry of the parabola. 

80 - Build up the theory of poles and polars for 

poles and polars. the parabola. 



CHAPTER XVI. 

ADDITIONAL WORK ON THE GENERAL EQUATION OF 
THE SECOND DEGREE. 

81. In Chapter XIII we investigated such 

necessity OF A properties of the curve represented by the 
general general equation of the second degree as 

treatment. would guide us in reducing the equation to 

certain standard or type forms. By the aid 
of these type forms any investigation which deals with the curve 
as an individual and without regard to its relative position may 
evidently be carried on. For example, the theorem that the locus 
of the points of intersection of perpendicular tangents to an ellipse 
is a circle concentric with the ellipse is altogether independent 
of the position of the ellipse, and therefore is proved with entire 
generality by the use of the type form. On the other hand, any 
theorem which depends upon the position of a conic with respect 
to the axes or to other curves cannot be established in its most 
general form by the use of type forms, since these presuppose a 
special position of the curve with respect to the axes. For 
example, the condition of tangency, equation of polar, values of 
sub-tangent and sub-normal hitherto deduced are all based upon 
the assumption that the curve occupies a particular position, and 
therefore they cannot be applied to the same curve in any other 
position. So long as we are dealing with a single curve there is 
no reason why we should not assume a particular position for it, 
but when two or more curves enter into the discussion only one 
of them in general can be given the desired position, and it is 
therefore necessary to make an investigation of the conic without 
regard to its position, i. e., to make a study of the general equation 
of the second degree. 

The results already secured are contained in a series of theorems 
in the opening paragraphs of Chapter XIII. 



110 Analytic Geometry. 

82 - When the left hand member of an equation 

degenerate consists of two or more rational factors, the 

CONICS. equation is satisfied by any set of values of 

the variables which make any one of the 
factors equal to zero. In consequence the locus corresponding to 
such an equation consists of two or more parts, which are them- 
selves the loci representing the equations formed by equating the 
various factors to zero. Such a locus is said to be degenerate. 
When a conic degenerates it is evident that the factors of the left 
hand member must be of the first degree and hence that the conic 
must degenerate into a pair of straight lines. The only questions 
of interest in such a case are concerning the point of intersection 
and the angle between the lines. The point of intersection 
evidently meets the definition of the center of symmetry and 
consequently may be found by solving the equations 

ax + hy + g = 

hx + l)y + f = 

To find the angle between the lines assume that the conic degener 
ates into the two lines 

a x x + Piij + y x = 
a 2 * + P»V + 7 2 = 0. 

Then the equation of the conic is 

a x a 2 x + P x P 2 y 2 + (a r ft + a.fijxy 

+ (a^ + a 2 yjx + (fty 2 + p a y x )y + yj* = 

Consider the two lines parallel to these lines and passing through 
the origin 

a,x + P x y = 

a. z x + P 2 y = 

These taken together constitute a degenerate conic 
a^ 4- ftfti/ + («A + a 2 PJxy = 

We may at once state the theorem that the second, degree part of 
the equation of a degenerate conic is the left hand member of the 
equation of a second degenerate conic whose lines intersect in the 
origin and are parallel to the lines of the first conic. 



Analytic Geometry. Ill 

It follows Unit if 

ax 2 + by 2 + 2hxy + 2gx + 2fy + c = 

represents a degenerate conic, the angle between the lines is equal 
to the angle between the lines of 

aw 2 + by 2 + 2hxy = 

This equation may be at once factored and the angle between 
the lines expressed by the relation 



a + b 

83 - It is desirable to have some test to apply 

discriminant. to a second degree equation in order to deter- 

mine at once whether or not it is degenerate. 
A moment's consideration will show that one peculiarity of a 
degenerate conic is that the center of symmetry, i. e., the point of 
intersection of the two component lines, is on the conic; and, 
conversely, the center of symmetry can be on the conic only when 
the conic degenerates. The necessary and sufficient condition for 
degeneracy of a conic is therefore the existence of a point which 
satisfies the three equations 

r ax + hy + g=0 

I hx + by + t = 
I ax 2 + by 2 + Zhxy + 2gx + 2fy + c = 0. 

The last of these may be written in the form 

x(ax + hy + g)+ y(hx + 1>y + fj+ gx + fy + e==0 

Equations I are therefore equivalent to 

ax + hy + g = 
hw + by+f=0 

g® + fy + c=o. 

But the necessary and sufficient condition for the simultaneous 
satisfaction of these three equations is 

a h g \ 

h I f\ = 0. 

9 f o \ 

This determinant, which plays an important part in the theory of 
conies, is called the discriminant of the equation of the conic. 
We shall denote it bv A. 



112 Analytic Geometry. 

84. After it has been shown that a conic is not 
classification. degenerate, the most important question is 

whether it is an ellipse, hyperbola, or para- 
bola. Article 72 and Problem 19, article 79, have shown that the 
ellipse has only imaginary points at infinity, that the hyperbola 
extends to infinity in two different directions, and that the two 
sides of the parabola tend to parallelism as the tracing point goes 
off to infinity. If then we join the infinite points on a conic to 
any point in the plane the resulting pair of lines will have 
imaginary slopes for the ellipse, real and equal slopes for the 
parabola, real and unequal slopes for the hyperbola. The lines 
which run from any point to the infinite points on the conic are 
the lines which give infinite values for r in the equation of article 
59, i. e., the lines whose slopes are such that we have 

al 2 -\-2hlm + l)m 2 = 0. 

The factors of this are imaginary, real and equal, or real and un- 
equal according as ab — h 2 is positive, zero, or negative. An equa- 
tion of the second degree in two variables therefore represents an 
ellipse, parabola, or hyperbola according as ab — h 2 is positive, 
zero, or negative. 

85. Since the tangent is a line meeting the 
TANGENTS AND conic in two coincident points, the condition 

that a line through (a? 1? y ± ) may be tangent 
to the conic represented by the general equa- 
tion of the second degree is that the two values of r given by 
the equation 

f(®i, V±) + 2r[l{aw 1 + hyi + g) + m{hx x + by x + f) ] 
+ r 2 (al 2 + 2hlm + bm 2 ) = 0* 

shall be equal. The condition for equal roots gives 

1 f( x i,yi){al 2 + 2fhlm + ~bm 2 ) 

—[l(ax 1 +hy 1 + g) + m(hx 1 + 6y 1 + f)I 2 = 0, 

an equation of the second degree in the ratio — whose two roots 

are the slopes of the two tangents from (a? x , y x ) to the conic. 



h /G»i, Vi) = «*\ + *y\ + 2^i2/i + 2gx!+ 2/yi + c. 



Analytic Geometry. 113 

Eliminating — between this equation and the equation of the line 

through (* u y x ) 

H x — x x _. y — y, 

I m 

we have 

III f(w n y 1 )[a(x — x 1 ) 2 -\-2h{x — x x )(y — y 1 )-{-b(y — y 1 )-] 
— [(x — x 1 )(ax 1 -^hy 1 + g)-\-(y — y 1 )(1ix 1 J r by 1 -\-f)y=i) 

as the equation of the pair of tangents from (x 1} y x ) to the conic.* 
If (#1? 2/i) is on the curve, f{x x , y x )=0, the two tangents coin- 
cide, the equation of the pair of tangents reduces to a perfect 
square, and we have as the equation of the tangent at the point 
(#1? Vi) on the conic 

{x — x r ) (wb 1 + hy 1 + g) + (y— y x ) (hx ± + by ± + f)=0. 

By transposing the negative terms to the right hand member and 
adding 

to both sides, this last equation is reduced to the form 

x(ax 1 + hy 1 + g) + y{hx 1 + by x + f) + gx x + fy ± + c = 
x 1 (ax 1 -\-hy 1 + g) + y 1 (hx 1 + by x + f) + gx x -\-fy 1 + c = 

f&i, yi)=o. 

we have therefore for the equation of the tangent at the point 
(#1? 2/i) on the curve 

x(ax 1 + %! + #) + ^(/^i + tyi + f) + 9%i + fVi+c = 0, 
which may also be written in the form 

x ± (ax + hy + g) + y x {l\® + by + f) + gx + fy + c = 0, 

as the student will see on multiplying out the two forms. 

The normal at any point on the conic is the perpendicular to 
the tangent at that point. With this definition the student should 
have no difficulty in writing its equation. 



*The significance of this elimination may not seem clear to the student. 
Equation II states that the point {x, y) is on a line through (x , y ) with 
direction cosines I, m. Equation I states that I and m are so determined 
that the line is tangent. Equation III, deduced by making I and II simul- 
taneous, states that {x, y) is on one or the other of the tangent lines. 



114 



Analytic Geometry. 



86. 
A SECOND 
CONDITION OF 
TANGENCY. 

first degree. Let 



The condition of tangency given in the last 
article is applicable when we have a line 
through a fixed point. We frequently need 
the condition of tangency applicable when 
the line is given by a general equation of the 





ax + Py + 1 

be a given straight line, what is the condition which must be sat- 
isfied in order that it ma}' be tangent to the conic 

ax 2 + ly 2 + 2hxy + 2gx-\-2fy + c = 0? 

Let us assume the line to be tangent at a point (x ly y x ). Its 
equation must then be 

x{ax 1 -f hy 1 + g) + y{lix x + by ± + f) + gx t + fy 1 + c = 0. 

This equation represents the same line as 

ax + Py + 1 = 
Therefore 

ax-, + Mix + ff — ^#i + h/i + ^ 0%i + /)/] + c 

a p 1 

For convenience of elimination equate each of these ratios to — n 
and we have 

ax 1 + hy 1 + g + =0 
lico 1 + by 1 + f-{-f3 l ^=0 
9^1 + fVi+c +m = 
ax, + Py, +1 = 

the last of which holds true because the point (x x , y t ) is on the 
line 

ax + Py + l = 0. 

The necessary and sufficient condition for the co-existence of these 
four equations is 

a h- g a 

h I f P 

9 f o 1 

a P 1 







which is therefore the condition which must be satisfied by the co- 
efficients of the given line and conic in order that they may be 
tangent to each other. 



Analytic Geometry. 115 

Expand this determinant, arrange the terms according to powers 
of a and /?, denote the minors of the determinant, taken with 
their proper signs, by A, B, C, F, G, H, as usual, and the condition 
of tangency takes the form 

Aa* + B(3 2 + 21Iap + 2Ga + 2F(3 + 0= 0* 

a form which we shall hereafter denote by 

From this general condition of tangency the student may at once 
deduce as special cases the conditions developed in articles 49, 70, 
and 79. 

What is the vital difference between the equation just developed 
and the equation of the conic? f(x, 2/) = is a condition which 
must be satisfied by the co-ordinates x and y of all points on the 
conic, i. e., the condition which selects from all the points in the 
plane those that lie on the conic. Similarly, F(a, /?) = is the 
condition which must be satisfied by the co-efficients of every line 
tangent to the conic, i. e., the condition which selects from all 
the lines in the plane those which are tangent to the conic. But 
a conic is just as fully determined by the aggregate of its tangents 
as by the aggregate of its points. The following questions im- 
mediately present themselves, (a) Since the co-efficients a and 
(3 determine the position of the line, are they not in some sense 
co-ordinates of the line? (b) Is not F(.a, P) = just as truly 
the equation of the conic as f (x, y) = 0? (c) Is it not possible to 
build up a geometry in which the variable element is the line 
rather than the point? (d) If so. would not the algebraic work 
of the two geometries be identical; and, therefore, could we not 
infer from each theorem already developed a new one differing 
from the old by an interchange of point and line? The answers 
to these questions lie beyond the scope of the present volume, but 
the student who will follow them up will find that they lead into 
one of the most interesting fields of modern mathematical investi- 
gation. 



""An immediate expansion of the determinant in this form is possible. 
Consult in any standard text on determinants the theorem on the expansion 
of a determinant in terms of the products in pairs of the constituents of 
any row and column. 



116 Analytic Geometry! 

87 - Any one of the methods used for deter- 

POLE AND POLAR, mining the equation of the polar of a point 
with respect to a conic might be extended 
to the general conic. We consider the one which regards the polar 
as the locus of harmonic conjugates of the pole with respect to the 
intersections of the conic and the chords through the pole. (This 
method was developed for the circle in article 55, and the notation 
and diagram of that article will serve for this also.) Let 

ax 2 + by 2 + 2hxy + 2gx + 2fy + c = 

be any conic, and (x x , y x ) any point P. Write the equations of 
any line through (x x , y x ) in the form 

® — x i + fr* 
y = y x -\-mr. 

Then the distances from the point {x ± , y ± ) to the conic, measured 
along this line, are given by the equation 

f{x x , y x ) +2r[l(ax 1 + hy, +g) + m{hx 1 -f by x + f) ] 

+ r 2 (al 2 + 6m 2 + 2hlm) = 
Denote these distances by r x and r 2 . Then 

1 | 1 _ 2l(ar; + Ji ih + g) + 2m(hx, + by, + f) 
i\ r 2 f(x 1} yj 

Let the harmonic conjugate, 8, of {x x , y x ) with respect to the two 
intersections of the chord and the conic be denoted by {x, y), 
then 

_^_ 2 . 

ps vV - x x y + w - Vl y 

We have also 

x — x x ?/' — Vx 

I = 77=, ., . , , ^ ^ 



V(x - x x J + (y - y x f V(x - x x Y + (y - y x Y 

The necessary and sufficient condition that 8 may be the harmonic 
conjugate of P with respect to the two intersections of the chord 
and the conic is 

>\ T, PS 

Substitute the values deduced above and reduce and we have, on 
dropping accents, 

x(ax t + %y ± + g) + y.{hx ± + by x +f) + g® 1 + fy 1 + c = 
which may also be written in the form 

x x {ax + hy + g) + y x (hx + by + f) + gx + fy + c = 0. 



Analytic Geometry. 117 

Either of these is therefore the equation of the polar of (x ly y t ) 
with respect to the conic f(x,y) — 0. 

The majority of the properties already established for poles and 
polars do not depend at all upon the choice of the system of 
reference, and therefore the demonstrations already given hold 
also for the case of the general conic. If the student is not satisfied 
as to the generality of any particular theorem, he should investi- 
gate the subject by the aid of the general equation just developed. 

88. Articles 61 to 63 enable us to find the 

length OF axes, equations of the axes of symmetry of any 
conic. This done, in any particular case it is 
theoretically an easy matter to find the distances between the 
intersections of these axes with the conic, i. e., the 
lengths of the axes. Practical difficulties of computation 
are apt to arise on account of the frequent presence of 
irrationals in the equations. These difficulties may be mini- 
mized by replacing the irrational by its decimal value carried to 
such a degree of approximation as the particular investigation 
may demand. It is also well to remember that what is needed is 
not the co-ordinates x x , y 19 x 2 , y 2 of the intersections, but the 
quantities (x ± — x 2 ) 2 and {y x — y 2 ) 2 ; and that if a and ft are the 
roots of 

ax 2 + 2bx + c = 0, 

4cb 2 — 4:ac * 



{a-PT 



a 



89- When the location of the center and the 
foci and lengths and directions of the axes of a conic 
eccentricity. have been determined, the location of the foci 

and the directrices and the determination of 
the eccentricity immediately follow.* 

90 - It has already been shown that the asymp- 
ASYMPTOTES. totes of any central conic 

ax 2 + /y = 1 

"An interesting treatment of this problem is given in C. Smith's Conic 
Sections, Article 171. 

fFor a proof of the existence of the four foci different from the one given 
in Article 66, see C. Smith, Conic Sections, Article 190. 



118 Analytic Geometry. 

referred to its axes of symmetry as axes of co-ordinates, are given 
by the two factors of 

ax 2 + /?// = 
If now we apply any transformation or series of transformations, 
of the sort hitherto considered, and thus transform the equation 
of the conic to the form 

I ax 2 + by 2 + 2hxy + 2gx + 2fy + c = 
the equation of the pair of asymptotes will become 
II ax 2 + by 2 + 2hxy + 2gx +2fy + k = 0, 

i. e., the equation of the pair of asymptotes of any central conic 
differs from the equation of the conic only in the constant term. 
(At first glance it seems as if c and k are connected by the re- 
lation, c=k — 1, but a moment's consideration will show that con- 
stant factors may have been introduced at any point in the trans- 
formation and that in consequence c and k may differ by any 
constant.) The determination of k presents, however, no serious 
difficulty. Equation II is an equation with one arbitrary para- 
meter, k, and therefore represents a family of conies distinguished 
from each other by the various values of k. In this family the 
pair of as3 r mptotes is included, and the value of k corresponding 
to this particular member of the family may be determined by 
giving algebraic expression to any one of its additional properties, 
just as in article 43 we determined a particular member of a 
family of lines by giving algebraic expression to some property of 
the line other than the one which it shared with all the members 
of the family. Now the conic whose equation we are seeking, the 
pair of asymptotes, is degenerate and this property of degeneracy 
is certainly not shared by all the conies of the family. If therefore 
We impose on the equation II the condition of degeneracy, we shall 
get the values of k corresponding to the degenerate members of 
the family; included among these will be the value of k corre- 
sponding to the pair of asymptotes. But the condition of degen- 
eracy is 

a h g 

h b f =0 

g f y t* . 

an equation of the first degree in k. There is therefore only one 



Analytic Geometry. 119 

degenerate member of the family;* and the value of k thus deter- 
mined gives us, when substituted in equation II, the equation of 
the pair of asymptotes of the conic represented by equation I. 

91 ■ If the conic represented by our general 

special equation of the second degree is in any par- 

treatment for ticular case a parabola, certain steps in our 
the parabola. general treatment (such as the determination 
of the center and the lengths of the axes) 
become impracticable, since they introduce infinite quantities into 
the discussion. What we actually need in order to determine the 
nature of a parabola from its equation is the location of its axis 
of symmetry and the tangent at its vertex, and the value of its 
parameter. The discussion to follow makes use of the following 
theorems : 

(A) The semi-parameter of a parabola is equal to the distance 
of any diameter from the axis of symmetry of the parabola, multi- 
plied by the tangent of the angle which the chords bisected by that 
diameter make with the axis of symmetry. (This theorem is a 
mere generalization of the solution of problem 1, article 78.) 

(B) Let 

S 1 = and S 2 = 

be any two straight lines. Then 

S x 2 + kS 2 =0 

is a conic passing through the intersection of these two lines, with 
$ 2 = as its tangent at this intersection. For if we substi- 
tute the value of y derived by solving S 2 = in the equation 
S x 2 -\- kS 2 = the resulting equation in a? is a perfect square, i. e., 
the line S 2 = meets the conic S ± 2 -\~ kS 2 = in two coincident 
points (is tangent), and since, as is easily seen, the co-ordinates of 
these points satisfy both S x = and S 2 = the point of tangency 
is the intersection of the two lines. 

Let it be granted that the general equation 

ax 2 + by 2 + 2hxy + 2gx + 2/=y + c = 

represents a parabola. Then we have 

ab — li 2 = 0, 

*That is, only one with a finite value of 7c. But the asymptotes of a 
central conic are a pair of lines in the finite part of the plane and hence 
all the coefficients in their equation are finite. 



120 Analytic Geometry. 

and the general equation of a diameter takes the form 

ax + Vab y + g + "7 (Vab x + by + f) = 

that is {a + ™Vab)x + (Va~b + jb) y + g + ™f=0 

(a result which gives us the theorem that all diameters of a para- 
bola are parallel) . For the axes of symmetry we must have 

lm\ 1 = Q 



. I 



I / i/ab \ I 
m 



-VW; 



Substituting the first of these values in the general equation of 
a diameter we have a constant equal to zero, i. e., one of the 
diameters of the parabola is the line at infinity, a fact which we 
already knew. Substituting the other value we have 

(a + b ) \/~a% + (a +b)^/by +y/ag +\Tbf= 

i. e., V ax + Vby + ^l^±^f-= 

a -j-o 

the equation of the axis of symmetry of the parabola. Since all 
diameters of the parabola are parallel, the particular one which 
passes through the origin is 

\/~ax + Yby = 

and the perpendicular distance from this diameter to the axis of 
symmetry is 

\/ag + ybf 

~ (a + bf 

The original equation may be put in the form 

( V ax + V 6 y ) 2 + 2gx + 2fy + c = 
Therefore 2gx + 2fy + c = 

is the tangent to the parabola at the point where it is met by 
the diameter through the origin. The slope of this tangent and 
therefore of the chords bisected by the diameter 

\/~ax-\- V% = 

is — 'JL and the tangent of the angle which these chords make 



Analytic Geometry. 
with the axis of symmetify is 



121 



therefore 



P = 



Vaf—ybg 
I &Q + Vbf 
Vaf— v Ig 
{a + &) 1 

The student has now sufficient material at hand to determine 
all the data concerning an}- particular parabola. 
Example. As an illustration consider the equation 

4a? 2 + ixy + if + 6a? + 2y + 4 = 0, 

i. e., (2# + 2/) 2 +*6# + 2i/ + 4 = 0. 

The axis of symmetry is 



2# + 2/+X=0 
o 



and the semi-parameter is 



25 l/o 

The yertex of the conic is its intersection with the axis of sym- 
metry, i. e., 

'9 4A\ 
50 



vy, 44 

50 25/ 

The tangent at the yertex is therefore 



and the directrix is parallel to this 



at a distance of 



50 



1/5. 



All that is needed to complete our 
information is to know which way 
the parabola is turned. This is 
settled at once by the co-ordinates 
of any other point on the conic. For 
example the diameter 

2a? + y = 

meets the curve at ( — 2, 4) which 
lies to the left of the tangent at the 
vertex. The curve is therefore as drawn. 




Fig. 32. 



122 Analytic Geometry. 

92 - The student has now at his disposal sufti- 

H IP H F R 1 nd 

■ cient formulae to enable him to determine 

the character, form, and location of the curve 
represented by any algebraic equation of the second degree. Loci 
corresponding to higher degree algebraic equations or to trans- 
cendental equations, as well as curves and surfaces in space, can 
be treated in a more satisfactory way after the student has ac- 
quired a knowledge of the elements of the differential and integral 
calculus. 

93 - PROBLEMS. 

FAMILIES OF CONICS. 

1. Given that 
^ = and S 2 = 
are the equations of two conies, show that 

# l + &&2 = 0, 

where k is an arbitrary constant, represents a family of conies, 
singly infinite in number, each of which passes through all four 
of the intersections of the two original conies. 

2. Five points on a conic are sufficient to determine it uniquely. 
Therefore among the conies through four given points, one and 
only one passes through each of the remaining points of the plane. 
Hence show that the family 

includes every possible conic through the intersections of 

& = 6 and S 2 — 0. 

.3. Given the two conies 

3% 2 + 2y 2 + 4,% — 1=0 
x 2 — y 2 + 2y — 3 = 

form the equation of the family of conies through the intersections 
of these two conies, and find the equation of that member of the 
family which passes through the origin. 

4. Show that the family of the last problem includes one circle, 
two parabolas, and three degenerate members. To what extent 
are these statements true of the general case of problem 1 ? 

5. How many members of the family in problem 1 are tangent 
to any given line? 



CHAPTER XVII. 

OTHER SYSTEMS OF CO-ORDINATES, 
POLAR CO-ORDINATES. 

94 - We must not assume that the system of co- 

various systems ordinates we have been using is the only 

OF co-ordinates, system in use. On the contrary any set of 

quantities which will serve to determine the 

position of a point may be taken as co-ordinates of the point. 

In the Cartesian system the co-ordinates of a point are the 
distances of two lines, x^a, y = 'b, from the base lines which 
form the system of reference. In other words the point is located 
as the intersection of two lines. 

Next in point of simplicity comes a system in which the point 
is located as the intersection of a straight line and a circle. In 
this the system of reference is a fixed 
point A and a fixed line AB. Any point 
P in the plane may now be located by 
giving p the radius of the circle cen- 
tered at A and passing through P, and 
the angle between the base line AB 
and the line AP. It is evident that in ? 
this system a pair of co-ordinates, p, 0, 
determines the point P not uniquely, 

. . , Fig. 33. 

but as one of two, i. e., either P or P . 

Next in order of simplicity comes a system of bi-polar co-ordi- 
nates in which the point is located as 
the intersection of two circles. The 
system of reference consists of two 
fixed points A and B and the two co- 
ordinates are the radii of two circles 
centered at A and B. It is evident that 
in this system a pair of co-ordinates 
does not determine a point uniquely, 
but as one of two, i. e., either P or P' . 






124 Analytic Geometry. 

Consider still another system. Take 
two ellipses, concentric and co-axial, 
and let the semi-axes of the one be a 
and 2a, and the semi-axes of the other 
2& and ~b. Then a and b are the co- 
ordinates of the intersections of the two 
ellipses. It is evident that any pair of 
co-ordinates does not determine a point 
uniquely, but as one of four, P, P' , P", 
P"\ 

In general given any two families of curves, each of which 
depends upon a single arbitrary parameter, and given also a point 
jP, we can determine the values of. the parameters giving in 
each family the particular member which passes through P. This 
pair of values may be regarded as the co-ordinates of P, deter- 
mining P, not uniquely, but as one of the intersections of the two 
curves given by the chosen values of the parameters. It is now 
evident that the number of systems of co-ordinates is infinite. 

95 - Certain systems of co-ordinates possess 

merits and particular merit for the investigation of some 

DEMERITS OF particular problem. In the bipolar for ex- 

various systems, ample, if we take the base points as the foci 
of an ellipse or hyperbola the equations of 
these curves reduce to the simple forms a? + y ■== h, and x — y = lc. 
On the other hand each system has certain demerits. In the bi- 
polar system for example, the simplicity of the form taken by the 
equations of certain conies is more than offset by the complexity 
of the equation of the straight line. 

In one particular all of the systems so far discussed are seriously 
lacking. Given two algebraic variables of the most general type, 
x = x 1 -\- ix 2 , and y==y ± -f- iy 2 , we can assign to x ly x 2 , y 1} y 2 any 
values whatever, and therefore can form a quadruply infinite 
number of pairs of values of x and y. But all these systems of 
co-ordinates attempt to represent pairs of values of x and y by 
points in the plane, and such points are only doubly infinite in 
number. In consequence each system must fail to give a complete 
geometric representation to the algebraic relations under con- 
sideration. Some of the systems, as may be seen from the prob- 
lems which follow this article, leave pairs of real values of the 



Analytic Geometry. 125 

variables without point representation, while others represent real 
points by pairs of imaginary values of the variables. 

The majority of the systems of co-ordinates which may be used 
fail also in another important particular in that they do not 
establish a one to one correspondence between points and pairs 
of values, but determine two or more points as corresponding to a 
single pair of values. One of the great beauties of the Cartesian- 
system is that it establishes a one to one correspondence between, 
the points of the plane and all pairs of real values of two algebraic 
variables. 

PROBLEMS. 

1. Given a system of bi-polar co-ordinates in which the distance 
between the base points is 8, plot the locus x — \j = 0. 

2. Is the point (3, 3) ou the above locus? locate it iu the 
diagram. 

3. Plot the ellipse x -f- y = 10 in the same system. 

1. Plot the hyperbola x — y = 10 in the same system. 

5. What conditions must be met by the co-ordinates of a point 
in order that it may be represented on the diagram if the distance 
between the base points is fc? 

6. Given the two families of ellipses whose equations in Carte- 
sian co-ordinates are 

x +-4 = 1, and -£+^ = 1 
a o 

then the values of a and ft determining the ellipses through any 
point are the co-ordinates of that point in the new system in which 
the ellipses are the determining elements. What are the Cartesian 
co-ordinates of the points whose co-ordinates in the new system 

are (4,2), (4,1), (1,1), (i,2)? 

7. Plot in this same system the locus for which a= o. 

8. What are the co-ordinates in this system of the point whose 
Cartesian co-ordinates are (3, 3) ? 

96. Polar co-ordinates are of particular value 

POLAR l n anY investigation in which the important 

CO-ORDINATES. . " , ° ' . , , ,. ,. 

elements are the distance and direction oi 
the variable point from a fixed point. The 

movement of the earth about the sun or any problem concerning a 

spiral curve are illustrations. 



126 Analytic Geometry. 

The drawbacks to ishe system as it was outlined in article ( .)4 are 
numerous. Given a pair of values p, ft we note first that while 
may have any value p must be positive and that the point is deter- 
mined only as one of two. On the other hand given a point ? 
we have for it a single value of p, but any number of values of 
i.e.,0,0+2Tr,0+4TT,0+Q>iT....0+2,nT. (Hereafter we shall call the? 
of any point the radius vector and the the amplitude of the 
point.) In other words any real amplitude may be paired with any 
real positive radius vector and the combination will be represented 
by either one of two points, while a point is represented by a real 
radius vector paired with any one of an infinity of amplitudes 
differing from each other by integer multiples of 2 77-. 

Mathematicians are accustomed however to make an assumption 
which enables them to include negative values of the radius vector. 
Abandoning the circle they fix their attention on the angle and 
the distance jo, and agree that the positive value-of p is to be meas- 
ured from the vertex of the angle in the direction of the boundary 
of and a negative value of p in the opposite direction. With this 
agreement the co-ordinates (p, 0) denote 
the point P and the co-ordinates ( — p, 0) 
the point P '. But the point P has now the 
co-ordinates (p, 0) or ( — p, + it), or 
more generally (p, + 2mt) or (— p, 
+ (2n + 1)77\). In other words any real 
pair of values of p and is represented 

-i . Fig. 36. 

by a single point while any point has 

two radii vectores and an infinity of amplitudes differing from 
each other by integer multiplies of it. 

PROBLEMS. 

Plot the curves corresponding to the following equations : 

1.9 = 4,. 2. 0=2. 

3. p= 0. 4. p = shift 

5. p = eP. 6. ft +30 +2 = 0. 

7. Show that the area of a triangle with one vertex at the pole 

(the fixed point) and the others at (p x , Qj and (/>,, ft) is — p 1 p* sin 

(ft — ft). 

8. Show that the area of the triangle whose vertices are (p n ft), 

(P., 2 ), 3 , ft) is \ {piP* sin (ft - ft) + PJ>* sin (ft - ft) 

+ Aft Bin (ft — ft)} 




Analytic Geometry. 



127 



97. 
THE RELATION OF 
POLAR AND 
CARTESIAN 
CO-ORDINATES. 

cides with the X axis, 
have at once 



Cartesian and polar co-ordinates are con- 
nected by simple relations which make the 
transformation from one to the other an easy 
matter. We consider first the case where the 
pole is at the origin and the base line coin- 
In this case we 

Y 



X == p cos 
y = p sin 6 



whence 



P = V x- + y\ 

= tan" 1 y . 

x 

When the pole is at the origin and 
the base line OA makes an angle a 
with the axis of X we have 



X 



Fig. 37. 



X = p COS (0 + a) 
y—p sin (0 + a) 



whence 




0= tan" 1 ^ — a 

Fig. 38. 

When the Cartesian system is not 
in either one of these positions it can easily be placed in one or 
the other by a movement of the axes parallel to themselves. There- 
fore in the most general case when the pole is at the point (a, J)) 
and the base line O'A makes an 
angle a with the axis of X we have 



whence 



x - 

y- 


-a=p 
-b= P 


cos (0 + a) 
sin (9 + a) 




p z 
6 = 


= V{x 
- tan -1 


- a7 + Cy - 

V~ l a 


by 



* 








p 






7 








i 









a 









x — a 

If it is desired to pass from one fig. 39. 

polar system to another whose pole 
is not coincident with the first it is simpler to transform first to a 



128 



Analytic Geometry. 



Cartesian system whose origin and X axis coincide with the pole 
and base line of the first system and then transform to the 
second polar system. 

PKOBLEMS. 



1. Show that the equation of the circle centered at ( / o 1 , ft) and 
of radius r is p 2 — 2 PPl cos (0 — ft) + p, 2 — r 2 = 0. 

2. Find the equation of a circle when the pole is on the circum- 
ference and the base line is the tangent at the pole. 

3. Find the equations of the various conic sections when the 
pole is at one focus and the base line is the axis of symmetry 
through that focus, and show that any one of them may be reduced 
to the form 



P 



— 1 — e' cos 



where p and e are the semi-parameter 
and the eccentricity. 

4. Let AB be the directrix, F the 
focus, and P any point on the conic. 
Deduce the equation just given directly 
from the diagram. ( It is sometimes £ 
convenient to measure the angle from 
FG as a base line. In that case the 
equation takes the form 



P 



= 1 + e cos 0). 



IB 




Fig. 40. 



APPENDIX A. 

INFINITIES OF VARIOUS ORDERS. 

The student must bear in mind the fact that he is using the 
word infinity in a technical sense differing somewhat from the 
ordinary literary and philosophical usage of the word. In its 
ordinary usage infinity denotes that which exceeds all limitations 
and therefore no comparisons between various infinities are pos- 
sible. In mathematical usage infinity denotes that which increases 
indefinitely, and it is evident that between two such quantities a 
perfectly definite comparison may be made. Consider some illus- 
trations for the sake of clearness. In each of the three expressions 

2^ + 1 x 2 + 1 x 2 4- 1 
x x x A 

let x increase indefinitely. As this happens both numerator and 
denominator of each fraction increase indefinitely, i. e., in tech- 
nical phrase, become infinite, but the three fractions respectively 
tend to 2, increase indefinitely (tend to infinity) , and tend to zero. 
In other words, in the first case while both numerator and denom- 
inator increase they remain easily comparable with each other ; in 
the second case the numerator becomes incomparably greater than 
the denominator, and in the third case the numerator becomes in- 
comparably less than the denominator. Mathematicians express 
all this by saying that in the first case numerator and denominator 
are infinities of the same order, that in the second case the num- 
erator is an infinity of higher order than the denominator, that in 
the third case the numerator is an infinity of lower order than the 
denominator. In general given any two infinities x and y, x is said 
to be of the same order, a higher order, or a lower order than y 

according as the ratio — tends to a finite quantity, infinity, or 

V 
zero. If it is desired to make a still more accurate distinction 
some one infinity (in general the one of lowest order among those 
under consideration) is called an infinity of the first order and its 



130 Analytic Geometry. 

square, cube, ith power called infinities of the second, third, ith 
order. The order of any other infinity is then determined by 
comparison with these powers of the chosen infinity. Stated alge- 
braicly, let z be an infinity of the first order and y any other 
infinity. Then y is said to be an infinity of the Mh order when 

the limit of \ as y and z tend to infinity is a finite quantity differ- 
ent from zero. 



APPENDIX B. 
FUNCTIONALITY. 

One variable is said to be a function of another when the two are 
so related that a change in the one produces a change in the other.* 
Thus the momentum of a moving body is a function both of its 
mass and of its velocity. Many other examples of functionality 
in nature may easily be given. 

Any equation between two variables defines either as a function 
of the other. For example 

x 2 4- y 2 = 4 

so connects x and y that no change can be made in either without 
affecting the other. Consequently whether we shall call x a func- 
tion of y or y a function of a? is a matter of purely arbitrary choice. 
If the equation be solved for either of the variables, the one for 
which it is solved is said to be an explicit function of the other. 
Otherwise it is called an implicit function. Thus in the example 
last given x is an implicit function of y, but solve the equation for 
x and we have 



x = dfc V 4 — y 2 

which defines x as an explicit function of y. 

When for any reason it becomes desirable to indicate the fact 
of functional relation without specifying its exact nature we use 
the form 

y = f(x) 

which is variously read y equals a function of x, \j equals the f 
function of x, or more simply y equals the f of x. When several 
functions enter into the discussion they are distinguished by the 
use of subscripts f lf f 2 , f 3 , etc., or by the use of other letters F, <£ 



*This definition of functionality is merely a working definition for present 
use. In some branches of the higher mathematics it becomes necessary to 
define more closely and to recognize distinctions which the student is not 
now prepared to consider. 



132 Analytic Geometry. 

if/ etc., in the place of f. When in any discussion the form of the 
function corresponding to any functional symbol, F(x) for ex- 
ample, has been defined, it is understood that this symbol shall 
continue to represent this same form throughout the discussion. 
For example if we have 

f(x) = x 2 + 2a? — a 
then 

f(y)=y 2 +2y—a 

f{c)=c 2 +2c—a 

/-(a + /?)=(a+/3) 2 +2(a + £)-a 



APPENDIX C. 
PERMISSIBLE OPERATIONS. 

In reducing equations to a simple form as in article 24, the 
student must see to it that the operations performed do not in any 
way change the character of the locus. The force of this remark 
is most clearly set forth by some illustrations. 

The equations 

2x + y = x — 4 
and a? + 2/ + 4 = 

are equivalent, i. e., represent the same locus, since every point 
which satisfies either one satisfies the other also. 
The equations 

x + y ^=0 

' J 3 

and 3# + 3i/ — 7 = 

are equivalent since every point which satisfies either satisfies 
the other also. 
The equations 

x 2 + xy — 2x = 
and x + y — 2 = 

are not equivalent since the first is satisfied by every point on the Y 
axis and the second is not. 
The equations 

x = y 
and x 2 =y 2 

are not equivalent since the latter is satisfied by points for which 
x = — y as well as by those for which x = y. 

A little consideration will show that the modifications to which 
we subject our equations may be reduced to two operations : trans- 
position, and the introduction and rejection of factors. Still 
further consideration will show that a transposition of terms can 



134 Analytic Geometry. 

in no way affect the locus, and that the introduction or rejection 
of a factor has no effect upon the locus, provided that the factor 
is of such a nature that it cannot vanish for any values of the 
variables. The student will do well at this time to read the articles 
in some standard algebra upon reversible operations. See in 
particular Fine's College Algebra. 




APPENDIX D. 
PROJECTION. 

Let P be any point and CD any plane. 
Let Q be any other point. Draw PQ 
and let R be the point in which PQ ex- 
tended meets CD. Then R is the pro- 
jection of Q on the plane CD from P, 
sometimes called the center of projec- 
tion. The projection of any geometric 
object or group of objects is the aggre- 
gate of the projections of all the points 
of the object or group of objects. The 

projection is sometimes spoken of as the shadow of the object cast 
on the plane CD by a light at P, and this description is satis- 
factory when the point P and the plane CD are on opposite sides 
of the object. When this is not the case, as for Q' above, it is 
necessary to fall back on the definition first given. If this point P 
removes indefinitely, the lines joining P to the various points of 
the object to be projected tend to parallelism, i. e., projection 
from an infinite distance is by parallel lines. If in addition P 
removes indefinitely in a direction perpendicular to the plane CD, 
the projecting lines tend to parallelism and perpendicularity to 
CD, i. e., the perpendicular projection from an infinite distance 
consists of the aggregate of the feet of the perpendiculars let fall 
from all the points of the object upon the plane of projection. 
In this case the projection is said to be orthogonal and is evidently 
the sort already presented to the student in his study of solid 
geometry. 

If all the points of the object to be projected are in one plane 
EF and the center of projection P is in the same plane it is evident 
that the projection will be wholly in that plane and will consist 
of the totality of points in which lines from P to every point of 
the object meet LM, the line of intersection of CD and EF. If P, 
remaining in the same plane, removes indefinitely in a direction 



136 Analytic Geometry. 

perpendicular to the line LM, the projection of the object is the 
totality of the feet of the perpendiculars let fall from every point 
of the object on LM. In this case the projection is again called 
orthogonal, and is evidently the sort already presented to the 
student in his study of plane geometry. Hereafter when no center 
P of projection is mentioned it is assumed that the projection 
is orthogonal. 

If the student has a clear understanding of the preceding state- 
ments, the following propositions need no proof. 

I. The projection of any length AB upon the straight line CD, 
which makes with AB an angle 0, is AB cos#. 

II. The projection of any contour 
is the sum of the projections of its 
component parts. Proj. PQRS = 
Proj. PQ + Proj. QR + Proj. RS. 

III. For purposes of projection a FlG ' 42, 

curved line may be regarded as the limiting form of a broken line 
as the number of its component parts increases indefinitely and 
each part tends to zero. 

When the contour turns back upon itself the question at once 
arises whether or not the double portion shall be twice counted. 
So far as anything we have so far indicated is concerned the 
answer is yes. There is however another point of view. It is 
evident that if PQ be regarded as the path of a moving point, the 





Fig. 43. 

projection of PQ on CD is the amount of movement of the point in 
the direction CD while the point goes from P to Q. Looking at 
the question from this point. of view, which is an important one in 
much mathematical work, it is evident that the total movement 
in the direction CD of the point which travels such a path as 
PQRST is the sum of the projections of PQ, QR, RS, minus the 
projection of ST., i. e., P'S' — T'S' = P'T'. 

The point of view just outlined is the one usually taken by 
mathematical writers, and in accordance with it they adopt the 
following conventions : 



Analytic Geometry. 137 

• 

I. In considering any contour determine an initial and a ter- 
minal point. 

II. The positive direction in any portion of a contour is the 
direction of motion of a point going from the initial to the ter- 
minal point. 

III. Assume one direction or the other along the line on which 
projection is made as positive. 

IV. The projection of any of the component lines of a contour- 
is equal to the length of the line multiplied by the cosine of the 
angle between the positive direction of the projected line and 
the positive direction of the line on which the projection is made. 

The student may now prove the following theorems : 

I. The projection of any contour is equal to the algebraic sum 
of the projections of its component parts. 

II. The projection of any closed contour is zero. 

III. The projections of any two contours having the same 
initial and terminal points are equal. 

IV. The resultant of any open contour is defined to be a line 
joining its initial point to its terminal point. Show that the pro- 
jection of any open contour is equal to the projection of its 
resultant. 



APPENDIX E. 
IMAGINARIES. 

In all work with complex numbers the student must be careful 
to keep in mind the fact that the term imaginary is used in a 
purely technical sense. In their earlier mathematical »use, the 
terms real and imaginary undoubtedly had the significance they 
now have in ordinary usage, but a clearer understanding of the 
nature of number has come with the years and we no longer regard 
V — 1 as imaginary in the literary sense of the word any more 
than we think of a fraction as a broken number. 

The whole matter will perhaps be clearer if we look at the 
nature of the various sorts of number which are sometimes 
grouped together as algebraic. Algebra recognizes six operations, 
three direct (addition, multiplication, involution), three inverse 
(subtraction, division, evolution). As material to which to apply 
these operations the world about us presents nothing but positive 
integers. There are in nature no negative numbers and no frac- 
tions. An object divided into two parts becomes two objects, and it 
is only by imagining an undivided object to be divided, or imagin- 
ing two objects to be united that we are able to talk of halves. 

If now we apply any or all of the direct algebraic operations 
to the positive integers we get nothing new; sums, products, and 
positive integer powers of positive integers are all positive in- 
tegers, a fact which we may express by the statement that the 
positive integers form a complete group out of which it is impos- 
sible to pass by means of the direct algebraic operations. If 
however we apply to these positive integers the inverse operation 
of subtraction we may as before get positive integers, but we get 
also a new sort of thing to which we give the name of negative 
integers. The equations 

5 — 6 = ^, 4 — 6 = * 2 , 3 — 6 = * 3 , etc., 

define a set of numbers t 1} t 2 , f 3 , of such a nature that when they 
are increased by 1, 2, 3, they become zero, i. e., they bear the same 



Analytic Geometry. 139 

relation to zero that zero bears to 1, 2, 3. It is also evident that 
i a + l = t 2 ,t 2 + 1=^,^+1=0, 

i. e., £ 3 , t 2 , t Xi differ from each other in the same way as any three 
consecutive integers. In other words the inverse operation of 
subtraction enables us to define a series of negative integers, each 
one symmetric with respect to zero to a corresponding positive 
integer, and forming with zero and the positive integers an un- 
limited sequence of numbers such that it is possible to pass from 
any one to those consecutive with it on either side by the addition 
or subtraction of unity. 

Xow while nature knows no such things as a negative number 
and laughs at the idea of continuing the process of subtraction 
after zero is reached, she nevertheless presents numerous illus- 
trations of sequences of numbers arranged symmetrically with 
respect to some neutral point. Distance above and below sea level, 
north and south of the equator, or east and west of some chosen 
meridian ; temperature above and below the freezing point ; assets 
and liabilities, profit and loss are examples of such sequences, any 
one of which may be used to illustrate the properties and laws of 
operation connected with negative numbers. 

If we continue to apply the inverse algebraic operations we 
derive other new numbers to which we give the names of fraction, 
irrational, imaginary ; and then find that with positive and negative 
integers, fractions, irrationals, and imaginaries the field is again 
closed, in that no finite number of algebraic operations direct or 
inverse can give us anything new. For each of these new things, 
with the exception of the imaginary, it is possible to find with but 
little difficulty excellent illustrations in nature, and thereby 
render clearer to our minds the laws under which we work with 
these new symbols. 

It is even possible to find in nature an illustration of imagin- 
aries. Let us adopt the usual convention and represent positive 
and negative numbers by points on a straight line. Consider any 

positive number a and multiply it by? ( = i/ — 1) four times in suc- 
cession, arriving thus at the numbers 

ia, — a, — ia, a. In other words, two ^ a 

multiplications by i have the same effect FlG * 44, 

upon a as if it had been rotated 180 degrees on a circle centered at 

and of radius a. while four multiplications are equivalent to a 



140 



Analytic Geometry. 



> 



rotation through 360 degrees. We may therefore, for purposes 
of illustration, regard multiplication by i as equivalent to a rota- 
tion of 90 degrees. From this point of view real and pure imag- 
inary numbers may be represented by points on two perpendicular 
lines as in the diagram. A complex num- 
ber, such as a-\-ib, may be represented by 
a point with an abscissa a and an ordinate 
h; and thus by utilizing all the points of 
the plane we can represent all values of a 
single complex variable. The assumptions - 
by which this method of representation is 
built up may seem arbitrary or artificial, 

but they. amount merely to a recognition of the direction as well 
as the distance of a point from the origin, and so differ in degree 
rather than in kind from the assumptions by Avhich we represented 
positive and negative numbers by points on 
a straight line. 

Given any complex number a + %b repre- 
sented by the point P, the angle is called 
the amplitude or argument of P; and p, 
always considered positive, is called the mod- 
ulus of P. Evidently 



Fig. 45. 




Fig. 46. 



a = p COS 
h = p sin 



P = + Va + h 2 
= tan" 1 - 



and 



ib = p (cos + i sin 0). 



The term modulus is frequently abbreviated to mod. or denoted by 
a pair of vertical lines including the quantity considered. Thus 



modulus (a + i&)=mod(a+ *&)== |« + #>| =+Va- 2 + & 2 . 

If a complex number is zero the real part and the coefficient of 
the imaginary part are both zero, and conversely. 

If a complex number is zero its modulus is zero and conversely. 

The sum of two complex numbers consists of the sum of the real 
parts plus i times the sum of the coefficients of the imaginary 
parts. This theorem may be demonstrated instantly by adding the 
two complex quantities in the form a + ib, c + id. Geometrically, 



Analytic Geometry. 



141 




Fig. 47. 



lot the point /' represent a + ib, and 
( t ) represent c.+ t'd. Draw from P a 
line PR equal and parallel to OQ. 
Then # represents a + ib + c -f *d and 

07^= |«.+ t6 +c+'id|. 
The justification of this geometric treat- 
ment of addition is left to the student. 
If he has ever studied physics, the idea 
Avill doubtless suggest itself to him that 
complex quantities might with profit be 
used in the discussion of such topics as 
the resolution and composition of forces or motions. 

The geometric treatment of addition leads at once to the import- 
ant theorem that the modulus of the sum of two (or any number 
of) complex numbers is less than, or at most equal to, the sum of 
the moduli. Is it possible to state a similar theorem concerning 
the amplitudes? 

To form the product v of two complex numbers a -f- ib, c + id 

let \a + ih\ = p, amp.(« + ib) = 0, \c + id\ = /*, amp. (c + id)= 4>, 
then (a + ib) (c + id) = p (cos + i sin 6) Kcos <£ + i sin 4>) 

= pp- (cos cos <£ — sin Q sin <p + i (cos sin <t> + sin cos <£)) 
= PP- (cos ( + <£) + i sin (0 + <£)) 
It follows at once that the modulus of the product of two complex 
numbers is the product of the moduli, and the amplitude of the 
product is the sum of the amplitudes. 

This last theorem leads to a simple geometric method of con- 
structing the point which represents the product of two complex 
numbers. Let P represent a +- ib = p (cos + i sin 0) and Q repre- 
sent c +- id = fi (cos <f> ■ +- i sin <j> ) and 
OA be the unit of measure on which 
the diagram is constructed. Construct 
the angle iff = + 4> and at Q construct 
the angle a = OAP. Then R, the in- 
tersection of the two lines so deter- 
mined, represents {a -+ ib) {c +- id) 
= pp. ( cos ( 9 + <£ ) + i sin (0 + 40) 
since its amplitude is the sum of the 

amplitudes and (since U t = — - — ' 

OA OQ 

i. e., OR = OP.OQ) its modulus is the product of the moduli. 




142 Analytic Geometry. 

The student who desires to make a further study of this mode 
of representing a complex variable may read, among others, 
Burnside & Panton, Theory of Equations, Vol. I, Chap. XII; or 
Durege, Elements of the Theory of Functions of a Complex 
Variable, Introduction. 



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